Probability Explanation And Exercises Worksheet Page 12
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37This problem is best approached by asking what is the probability that no
two people have the same birthday. Once we know this probability, we can simply
subtract it from 1 to find the probability that two people share a birthday.
If we choose two people at random, what is the probability that they do not
share a birthday? Of the 365 days on which the second person could have a
birthday, 364 of them are different from the first person's birthday. Therefore the
probability is 364/365. Let's define P2 as the probability that the second person
drawn does not share a birthday with the person drawn previously. P2 is therefore
364/365. Now define P3 as the probability that the third person drawn does not
share a birthday with anyone drawn previously given that there are no previous
birthday matches. P3 is therefore a conditional probability. If there are no previous
birthday matches, then two of the 365 days have been “used up,” leaving 363 non-
matching days. Therefore P3 = 363/365. In like manner, P4 = 362/365, P5 =
361/365, and so on up to P25 = 341/365.
In order for there to be no matches, the second person must not match any
previous person and the third person must not match any previous person, and the
fourth person must not match any previous person, etc. Since P(A and B) =
P(A)P(B), all we have to do is multiply P2, P3, P4 ...P25 together. The result is
0.431. Therefore the probability of at least one match is 0.569.
Gambler’s Fallacy
A fair coin is flipped five times and comes up heads each time. What is the
probability that it will come up heads on the sixth flip? The correct answer is, of
course, 1/2. But many people believe that a tail is more likely to occur after
throwing five heads. Their
faulty reasoning
may go something like this: “In the
long run, the number of heads and tails will be the same, so the tails have some
catching up to do.”
The error in this reasoning is that the proportion of heads approaches 0.5 but
the number of heads does not approach the number of tails. The results of a
simulation
(external
link; requires Java) are shown in Figure 1. (The quality of the
image is somewhat low because it was captured from the screen.)
196
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education