Week 5 Worksheet 11 MATH 10B
Tu 2/28/19
1. Concept of Independence
•What is the probability that two people chosen at random were born during the
same month of the year?
Solution: 12
11
12
1
12 =1
12
•What is the probability that in a group of n people chosen at random, there are
at least two born in the same month of the year?
Solution: If n > 12, by the pigeonhole principle, at least two people were
born in the same month. Otherwise, P(at least two born in the same month) =
1−P(all born in different month) = 1 −12
12 ×11
12 ×10
12 . . . 12−n+1
12 .
•How many people chosen at random are needed to make the probability greater
than 1/2 that there are at least two people born in the same month of the year?
Solution: In other words, we want to find the minimum nsuch that 1 −
P(all born in different month) = 1 −12
12 ×11
12 ×10
12 . . . 12−n+1
12 >1
2. When
n= 4, P (at least two born in the same month) ≈0.482, and When n=
5, P (at least two born in the same month) ≈0.618. Thus, n≥5.
2. Two dies were rolled. Are the events that the first die rolled is a 1 and that the sum
of the two dice is a 7 independent?
Solution: Let Abe the event that first die rolled is a 1, and Bbe the event that
the sum of the two dice is a 7. Then,
P(A∩B) = 1
36 (1)
P(A) = 1
6(2)
P(B) = 6
36 (3)
Therefore, P(A∩B) = P(A)P(B).
3. Find the smallest number of people you need to choose at random so that the proba-
bility that at least two of them were both born on April 1st exceeds 1/2.
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