In the previous section we saw the results when the number of times a die is rolled is increased. In this section we will see such experiments related to 'tossing two coins simultaneously'.
Let us do an experiment. We will call it experiment IIIA:
1. Take two coins.
(i) Toss them simultaneously. The reading may be any one of the following:
HH, TT and HT
Note that, in the result HT, it does not matter which coin gives H and which one T
(ii) What ever be the reading, note that reading on a piece of paper.
• (i) and (ii) constitutes one cycle of our experiment.
2. Repeat the cycle 10 times.
3. In the note book, tabulate the readings as shown in table 28.5 below
Table 28.5
4. Determine the following ratios:
• Number of times no head turned up⁄Total number of times the two coins are tossed
• Number of times one head turned up⁄Total number of times the two coins are tossed
• Number of times two heads turned up⁄Total number of times the two coins are tossed
• In our present case:
♦ the 1st ratio is 2⁄10 = 0.2
♦ the 2nd ratio is 6⁄10 = 0.6
♦ the 3rd ratio is 1⁄10 = 0.1
5. Once the ratios in (4) are determined, the experiment IIIA is complete
• For this experiment IIIB, the number of cycles in (2) must be 20
• Let the readings be as shown below:
• In this case:
♦ the 1st ratio is 5⁄20 = 0.25
♦ the 2nd ratio is 11⁄20 = 0.55
♦ the 3rd ratio is 4⁄20 = 0.2
• When the ratios in (4) are determined, the experiment IIIB is over
■ Once again repeat the experiment. We will call it experiment IIIC
• For this experiment IIIC, the number of cycles in (2) must be 30
• Let the readings be as shown below:
• In this case:
♦ the 1st ratio is 8⁄30 = 0.267
♦ the 2nd ratio is 17⁄30 = 0.567
♦ the 3rd ratio is 5⁄30 = 0.167
• When the ratios in (4) are determined, the experiment IIIC is over
So we did the same experiment 3 times. Before proceeding further, we will discuss the importance of the six ratios:
♦ Number of times TT is obtained⁄n
♦ Number of times HT is obtained⁄n
♦ Number of times HH is obtained⁄n
■ Let us now analyse the ratios:
1st ratio when number of trials is 10 = 0.2
1st ratio when number of trials is 20 = 0.25
1st ratio when number of trials is 30 = 0.267
■ As the number of trial increases, the 1st ratio gets closer and closer to 0.25
Consider the 2nd ratio:
2nd ratio when number of trials is 10 = 0.6
2nd ratio when number of trials is 20 = 0.55
2nd ratio when number of trials is 30 = 0.567
■ As the number of trial increases, the 2nd ratio gets closer and closer to 0.5
Consider the 3rd ratio:
3rd ratio when number of trials is 10 = 0.1
3rd ratio when number of trials is 20 = 0.2
3rd ratio when number of trials is 30 = 0.167
■ As the number of trial increases, the 3rd ratio gets closer and closer to 0.25
• We can increase the 'number of trials' to a 'considerably large value' by using groups as we saw in the previous sections. Students may try it themselves.
• It will become clear that the ratios get closer and closer to 0.25, 0.5 and 0.25
Let us do an experiment. We will call it experiment IIIA:
1. Take two coins.
(i) Toss them simultaneously. The reading may be any one of the following:
HH, TT and HT
Note that, in the result HT, it does not matter which coin gives H and which one T
(ii) What ever be the reading, note that reading on a piece of paper.
• (i) and (ii) constitutes one cycle of our experiment.
2. Repeat the cycle 10 times.
3. In the note book, tabulate the readings as shown in table 28.5 below
Table 28.5
| Number of times the two coins are tossed | Number of times no heads comes up (TT) | Number of times one head comes up (HT) | Number of times two heads come up (HH) |
|---|---|---|---|
| 10 | 2 | 6 | 1 |
• Number of times no head turned up⁄Total number of times the two coins are tossed
• Number of times one head turned up⁄Total number of times the two coins are tossed
• Number of times two heads turned up⁄Total number of times the two coins are tossed
• In our present case:
♦ the 1st ratio is 2⁄10 = 0.2
♦ the 2nd ratio is 6⁄10 = 0.6
♦ the 3rd ratio is 1⁄10 = 0.1
5. Once the ratios in (4) are determined, the experiment IIIA is complete
But our work is not over
■ Repeat the above experiment. We will call it experiment IIIB.• For this experiment IIIB, the number of cycles in (2) must be 20
• Let the readings be as shown below:
| Number of times the two coins are tossed | Number of times no heads comes up (TT) | Number of times one head comes up (HT) | Number of times two heads come up (HH) |
|---|---|---|---|
| 20 | 5 | 11 | 4 |
♦ the 1st ratio is 5⁄20 = 0.25
♦ the 2nd ratio is 11⁄20 = 0.55
♦ the 3rd ratio is 4⁄20 = 0.2
• When the ratios in (4) are determined, the experiment IIIB is over
■ Once again repeat the experiment. We will call it experiment IIIC
• For this experiment IIIC, the number of cycles in (2) must be 30
• Let the readings be as shown below:
| Number of times the two coins are tossed | Number of times no heads comes up (TT) | Number of times one head comes up (HT) | Number of times two heads come up (HH) |
|---|---|---|---|
| 30 | 8 | 17 | 5 |
♦ the 1st ratio is 8⁄30 = 0.267
♦ the 2nd ratio is 17⁄30 = 0.567
♦ the 3rd ratio is 5⁄30 = 0.167
• When the ratios in (4) are determined, the experiment IIIC is over
1. We know that, the probability of obtaining TT, HT and HH when tossing two coins simultaneously are 0.25, 0.5 and 0.25 respectively
2. But these are theoretical values. If they are always obtained in the real life also, we will get results such as these:
• Toss two coins simultaneously 12 times
♦ TT will be obtained 3 times
♦ HT will be obtained 6 times
♦ HH will be obtained 3 times
• Toss two coins simultaneously 16 times
♦ TT will be obtained 4 times
♦ HT will be obtained 8 times
♦ HH will be obtained 4 times
♦ TT will be obtained 3 times
♦ HT will be obtained 6 times
♦ HH will be obtained 3 times
• Toss two coins simultaneously 16 times
♦ TT will be obtained 4 times
♦ HT will be obtained 8 times
♦ HH will be obtained 4 times
3. But we never get such exact values.
4. However, as the number of trials increase, each of the 3 ratios become closer and closer to their respective values 0.25, 0.5 and 0.25
4. However, as the number of trials increase, each of the 3 ratios become closer and closer to their respective values 0.25, 0.5 and 0.25
5. If, instead of 10,20 or 30 times, if we toss it for a 'very large number of times, n', then:
• Number of times TT is obtained⁄n = 0.25
• Number of times HT is obtained⁄n = 0.5
• Number of times HH is obtained⁄n = 0.25
• Number of times TT is obtained⁄n = 0.25
• Number of times HT is obtained⁄n = 0.5
• Number of times HH is obtained⁄n = 0.25
We are trying to prove this using our present experiments
• From the three experiments, we have three sets of ratios. One set from each experiment
• Each set has six ratios:♦ Number of times TT is obtained⁄n
♦ Number of times HT is obtained⁄n
♦ Number of times HH is obtained⁄n
■ Let us now analyse the ratios:
1st ratio when number of trials is 10 = 0.2
1st ratio when number of trials is 20 = 0.25
1st ratio when number of trials is 30 = 0.267
■ As the number of trial increases, the 1st ratio gets closer and closer to 0.25
Consider the 2nd ratio:
2nd ratio when number of trials is 10 = 0.6
2nd ratio when number of trials is 20 = 0.55
2nd ratio when number of trials is 30 = 0.567
■ As the number of trial increases, the 2nd ratio gets closer and closer to 0.5
Consider the 3rd ratio:
3rd ratio when number of trials is 10 = 0.1
3rd ratio when number of trials is 20 = 0.2
3rd ratio when number of trials is 30 = 0.167
■ As the number of trial increases, the 3rd ratio gets closer and closer to 0.25
• It will become clear that the ratios get closer and closer to 0.25, 0.5 and 0.25
In the next section, we will see how the above findings can be used to solve practical problems.
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