In the previous section we completed the discussion on Pie charts. In this section we will discuss about Probability.
In our day to day life, we use words related to 'probability' many times. Some examples are:
• There is a high probability that a question may be asked from this particular topic in the examination.
• Probably it will rain today. The sky is a bit cloudy
• It is less probable that we will get the car serviced by today evening as they had promised. Some workers are on leave.
In the above statements, we notice that there is an element of chance. An event may or may not occur. In those situations, one can not say for certain that an event ‘will’ occur. Neither can he say that an event ‘will not’ occur.
• A topic may be important. But the question sheet may not contain any questions from it.
• It may appear that the ‘rain will soon start’ on a cloudy day. But a wind may blow away the clouds, and it will become a sunny day. On the other hand, the wind can bring in more clouds and it may soon begin to rain.
• The car may not be ready, as some workers are on leave. But other workers might do ‘over time jobs’, and the car might be ready in time.
So we see that there is probability for an occurrence and non-occurrence. In the topic of ‘probability’, we try to predict an occurrence or non-occurrence by making simple mathematical models. First we will see the ‘degree of probability’.
On a very cloudy day there is a high probability that it will rain. The chance of a wind blowing away the clouds is low. So
• The degree of probability is high for one event. → The occurrence of rain
Here is another example:
A bag has 2 red balls and 5 blue balls. All the balls are identical except the colours. The balls are well shuffled and one ball is drawn with out looking. There is a high probability that, the drawn ball will be blue because the number of blue balls is greater. The chance for the drawn ball to be red is low. Then
• The degree of probability is high for one event. → The drawn ball to be blue.
In the two examples that we saw above, one event has a high degree of probability. So that event has an upper hand. For us, the other event, which is less probable, is inferior (even though, breaking all expectations, the inferior event ‘can’ indeed occur). Our aim is to show or present this high degree in a mathematical way.
But there are some situations where no event is inferior. All the possible events have equal chances of occurrence. Some examples are:
Tossing of a coin. Here, the two possible events are:
• The coin lands with Heads on the upper face
• The coin lands with Tails on the upper face
Both these two events have equal probability. None of them have an upper hand.
Rolling an unbiased cubical die. It has 6 numbers, from 1 to 6, with one number marked on each side. There are six possible events. They are: The die lands with
• 1 on the upper face
• 2 on the upper face
• 3 on the upper face
• 4 on the upper face
• 5 on the upper face
• 6 on the upper face
Each of the above six events have equal probability. None of them have an upper hand on the others because, it is an unbiased die. We must be able to present such equal probability also in a mathematical way.
Let us see how the mathematical formulation can be done. We will consider an experiment with the bag of balls that was mentioned earlier. The bag contains 2 red balls and 5 blue balls. All the balls are identical except the colours. Let us mark the two red balls as R1 and R2. And the 5 blue balls as B1, B2, B3, B4 and B5. When a person draws a ball with out looking, any one of the results (or outcomes) given below will occur:
• Outcome 1: The drawn ball is R1
• Outcome 2:The drawn ball is R2
• Outcome 3:The drawn ball is B1
• Outcome 4:The drawn ball is B2
• Outcome 5:The drawn ball is B3
• Outcome 6:The drawn ball is B4
• Outcome 7:The drawn ball is B5
● No outcomes other than the above 7 can possibly occur. So we say that the maximum number of outcomes possible is equal to 7.
● We want to present the high probability for the ‘occurrence of drawing a blue ball’. If a blue ball is drawn, we will call it an event. Let us examine each outcome:
Outcome 1: The drawn ball is R1 → Not a favourable outcome → We don’t have an event
Outcome 2: The drawn ball is R2 → Not a favourable outcome → We don’t have an event
Outcome 3: The drawn ball is B1 → A favourable outcome → We have an event
Outcome 4: The drawn ball is B2 → A favourable outcome → We have an event
Outcome 5: The drawn ball is B3 → A favourable outcome → We have an event
Outcome 6: The drawn ball is B4 → A favourable outcome → We have an event
Outcome 7: The drawn ball is B5 → A favourable outcome → We have an event
So, out of the 7 possible outcomes, 5 are favourable, and gives us an event. So,the probability for the occurance of the event (which is the ‘drawing of a blue ball’) is 5 in 7. In mathematical form, we write: The probability for the event of drawing a blue ball = 5⁄7
We want to know the probability for the red ball also. Then, drawing of a red ball is the event. The steps can be written as:
Outcome 1: The drawn ball is R1 → A favourable outcome → We have an event
Outcome 2: The drawn ball is R2 → A favourable outcome → We have an event
Outcome 3: The drawn ball is B1 → Not a favourable outcome → We don’t have an event
Outcome 4: The drawn ball is B2 → Not a favourable outcome → We don’t have an event
Outcome 5: The drawn ball is B3 → Not a favourable outcome → We don’t have an event
Outcome 6: The drawn ball is B4 → Not a favourable outcome → We don’t have an event
Outcome 7: The drawn ball is B5 → Not a favourable outcome → We don’t have an event
So, out of the 7 possible outcomes, 2 are favourable, and gives us an event. So,the probability for the occurance of the event (which is the ‘drawing of a red ball’) is 2 in 7. In mathematical form, we write: The probability for the event of drawing a red ball = 2⁄7
Now, we compare the quantities that we derived:
5⁄7 is greater than 2⁄7 . That is., 5⁄7 > 2⁄7.
The comparison can be presented in the form of a pie chart also. We learned how to draw pie chart in the previous section. For our present case, it will be as shown in the fig.1.38 below:
Thus, the high probability of the blue ball is proved.
[In the above experiment, it is important to note that the experiment is done only once. That is., the ball is drawn only once. We wrote a large number of sequential steps. Each of those steps is a ‘possible’ outcome in that single experiment. It should not be misunderstood that each outcome is an experiment.]
Now let us consider the experiment of tossing a coin. We want the probability for Heads. [This experiment is also done only once. That is., the coin is tossed only once] The possible outcomes are:
Outcome 1: The coin lands with Heads on the upper face.
Outcome 2: The coin lands with Tails on the upper face.
● No outcomes other than the above 2 can possibly occur. So we say that the maximum number of outcomes possible is equal to 2.
● We want to present the probability for ‘getting heads’. If we get heads, we will call it an event. Let us examine each outcome:
Outcome 1: The coin lands with Heads on the upper face. → A favourable outcome → We have an event.
Outcome 2: The coin lands with Tails on the upper face. → Not a favourable outcome → We don’t have an event.
So, out of the 2 possible outcomes, 1 is favourable, and gives us an event. So,the probability for the occurance of the event (which is ‘getting heads’) is 1 in 2. In mathematical form, we write: The probability for the event of getting heads = 1⁄2
We want to know the probability for the tails also. Then, getting tails is the event. The steps can be written as:
Outcome 1: The coin lands with Heads on the upper face. Not a favourable outcome. We don’t have an event.
Outcome 2: The coin lands with Tails on the upper face. A favourable outcome. We have an event.
So, out of the 2 possible outcomes, 1 is favourable, and gives us an event. So,the probability for the occurance of the event (which is ‘getting tails’) is 1 in 2. In mathematical form, we write: The probability for the event of getting tails = 1⁄2
Now, we compare the quantities that we derived:
Both are equal to 1⁄2. So the equal probability for heads and tails is proved. The presentation in the form of a pie chart is given below:
In the next section, we will consider the rolling of a die.
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In our day to day life, we use words related to 'probability' many times. Some examples are:
• There is a high probability that a question may be asked from this particular topic in the examination.
• Probably it will rain today. The sky is a bit cloudy
• It is less probable that we will get the car serviced by today evening as they had promised. Some workers are on leave.
In the above statements, we notice that there is an element of chance. An event may or may not occur. In those situations, one can not say for certain that an event ‘will’ occur. Neither can he say that an event ‘will not’ occur.
• A topic may be important. But the question sheet may not contain any questions from it.
• It may appear that the ‘rain will soon start’ on a cloudy day. But a wind may blow away the clouds, and it will become a sunny day. On the other hand, the wind can bring in more clouds and it may soon begin to rain.
• The car may not be ready, as some workers are on leave. But other workers might do ‘over time jobs’, and the car might be ready in time.
So we see that there is probability for an occurrence and non-occurrence. In the topic of ‘probability’, we try to predict an occurrence or non-occurrence by making simple mathematical models. First we will see the ‘degree of probability’.
On a very cloudy day there is a high probability that it will rain. The chance of a wind blowing away the clouds is low. So
• The degree of probability is high for one event. → The occurrence of rain
Here is another example:
A bag has 2 red balls and 5 blue balls. All the balls are identical except the colours. The balls are well shuffled and one ball is drawn with out looking. There is a high probability that, the drawn ball will be blue because the number of blue balls is greater. The chance for the drawn ball to be red is low. Then
• The degree of probability is high for one event. → The drawn ball to be blue.
In the two examples that we saw above, one event has a high degree of probability. So that event has an upper hand. For us, the other event, which is less probable, is inferior (even though, breaking all expectations, the inferior event ‘can’ indeed occur). Our aim is to show or present this high degree in a mathematical way.
But there are some situations where no event is inferior. All the possible events have equal chances of occurrence. Some examples are:
Tossing of a coin. Here, the two possible events are:
• The coin lands with Heads on the upper face
• The coin lands with Tails on the upper face
Both these two events have equal probability. None of them have an upper hand.
Rolling an unbiased cubical die. It has 6 numbers, from 1 to 6, with one number marked on each side. There are six possible events. They are: The die lands with
• 1 on the upper face
• 2 on the upper face
• 3 on the upper face
• 4 on the upper face
• 5 on the upper face
• 6 on the upper face
Each of the above six events have equal probability. None of them have an upper hand on the others because, it is an unbiased die. We must be able to present such equal probability also in a mathematical way.
Let us see how the mathematical formulation can be done. We will consider an experiment with the bag of balls that was mentioned earlier. The bag contains 2 red balls and 5 blue balls. All the balls are identical except the colours. Let us mark the two red balls as R1 and R2. And the 5 blue balls as B1, B2, B3, B4 and B5. When a person draws a ball with out looking, any one of the results (or outcomes) given below will occur:
• Outcome 1: The drawn ball is R1
• Outcome 2:The drawn ball is R2
• Outcome 3:The drawn ball is B1
• Outcome 4:The drawn ball is B2
• Outcome 5:The drawn ball is B3
• Outcome 6:The drawn ball is B4
• Outcome 7:The drawn ball is B5
● No outcomes other than the above 7 can possibly occur. So we say that the maximum number of outcomes possible is equal to 7.
● We want to present the high probability for the ‘occurrence of drawing a blue ball’. If a blue ball is drawn, we will call it an event. Let us examine each outcome:
Outcome 1: The drawn ball is R1 → Not a favourable outcome → We don’t have an event
Outcome 2: The drawn ball is R2 → Not a favourable outcome → We don’t have an event
Outcome 3: The drawn ball is B1 → A favourable outcome → We have an event
Outcome 4: The drawn ball is B2 → A favourable outcome → We have an event
Outcome 5: The drawn ball is B3 → A favourable outcome → We have an event
Outcome 6: The drawn ball is B4 → A favourable outcome → We have an event
Outcome 7: The drawn ball is B5 → A favourable outcome → We have an event
So, out of the 7 possible outcomes, 5 are favourable, and gives us an event. So,the probability for the occurance of the event (which is the ‘drawing of a blue ball’) is 5 in 7. In mathematical form, we write: The probability for the event of drawing a blue ball = 5⁄7
Outcome 1: The drawn ball is R1 → A favourable outcome → We have an event
Outcome 2: The drawn ball is R2 → A favourable outcome → We have an event
Outcome 3: The drawn ball is B1 → Not a favourable outcome → We don’t have an event
Outcome 4: The drawn ball is B2 → Not a favourable outcome → We don’t have an event
Outcome 5: The drawn ball is B3 → Not a favourable outcome → We don’t have an event
Outcome 6: The drawn ball is B4 → Not a favourable outcome → We don’t have an event
Outcome 7: The drawn ball is B5 → Not a favourable outcome → We don’t have an event
So, out of the 7 possible outcomes, 2 are favourable, and gives us an event. So,the probability for the occurance of the event (which is the ‘drawing of a red ball’) is 2 in 7. In mathematical form, we write: The probability for the event of drawing a red ball = 2⁄7
Now, we compare the quantities that we derived:
5⁄7 is greater than 2⁄7 . That is., 5⁄7 > 2⁄7.
The comparison can be presented in the form of a pie chart also. We learned how to draw pie chart in the previous section. For our present case, it will be as shown in the fig.1.38 below:
 |
| Fig.1.38 Probability for Red and blue ball |
Thus, the high probability of the blue ball is proved.
[In the above experiment, it is important to note that the experiment is done only once. That is., the ball is drawn only once. We wrote a large number of sequential steps. Each of those steps is a ‘possible’ outcome in that single experiment. It should not be misunderstood that each outcome is an experiment.]
Now let us consider the experiment of tossing a coin. We want the probability for Heads. [This experiment is also done only once. That is., the coin is tossed only once] The possible outcomes are:
Outcome 1: The coin lands with Heads on the upper face.
Outcome 2: The coin lands with Tails on the upper face.
● No outcomes other than the above 2 can possibly occur. So we say that the maximum number of outcomes possible is equal to 2.
● We want to present the probability for ‘getting heads’. If we get heads, we will call it an event. Let us examine each outcome:
Outcome 1: The coin lands with Heads on the upper face. → A favourable outcome → We have an event.
Outcome 2: The coin lands with Tails on the upper face. → Not a favourable outcome → We don’t have an event.
So, out of the 2 possible outcomes, 1 is favourable, and gives us an event. So,the probability for the occurance of the event (which is ‘getting heads’) is 1 in 2. In mathematical form, we write: The probability for the event of getting heads = 1⁄2
We want to know the probability for the tails also. Then, getting tails is the event. The steps can be written as:
Outcome 1: The coin lands with Heads on the upper face. Not a favourable outcome. We don’t have an event.
Outcome 2: The coin lands with Tails on the upper face. A favourable outcome. We have an event.
So, out of the 2 possible outcomes, 1 is favourable, and gives us an event. So,the probability for the occurance of the event (which is ‘getting tails’) is 1 in 2. In mathematical form, we write: The probability for the event of getting tails = 1⁄2
Now, we compare the quantities that we derived:
Both are equal to 1⁄2. So the equal probability for heads and tails is proved. The presentation in the form of a pie chart is given below:
 |
| Fig.1.39 Probability for Heads or Tails |
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