Question

It is known that 52% of the population participates in sports on a regular basis. Five random individuals are interviewed and asked whether they participate in sport on a regular basis. Let X be the number of people who regularly participate in a sport:

a) construct a probability distribution table for x

b) find the probability that 3 people or less play sports

c) Find the probability that at least one person plays a sport, given that no more than 3 people play a sport

d) Find the probability that the fist interviewed plays a sport but the next two do not

Answer 1

a)

Using Binomial distribution, X ~ Binomial(n=5, p = 0.52)

x	f(x)
0	5C0 * 0.520 * (1 - 0.52)5-0 = 0.0254804
1	5C1 * 0.521* (1 - 0.52)5-1 = 0.1380188
2	5C2 * 0.522 * (1 - 0.52)5-2 = 0.2990408
3	5C3 * 0.523 * (1 - 0.52)5-3 = 0.3239608
4	5C4 * 0.524 * (1 - 0.52)5-4 = 0.1754788
5	5C5 * 0.525 * (1 - 0.52)5-5 = 0.0380204

b)

Probability that 3 people or less play sports = P(X 3) = f(0) + f(1) + f(2) + f(3)

= 0.0254804 + 0.1380188 + 0.2990408 + 0.3239608

= 0.7865008

c)

Probability that at least one person plays a sport, given that no more than 3 people play a sport

= P(X 1 | X 3)

= P(X 1 and X 3) / P(X 3)

= P(1 X 3) / P(X 3)

= (f(1) + f(2) + f(3)) / 0.7865008

= (0.1380188 + 0.2990408 + 0.3239608) / 0.7865008

= 0.9676028

d)

Probability that the fist interviewed plays a sport but the next two do not = 0.52 * (1 - 0.52)²

= 0.119808