Question

For the random variables described below, which have binomial distributions?

1. You toss a balanced coin 50 times and count the number of heads.

2. Replacement heart valves have a probability of 0.77 of performing well for 15 years. It is reasonable to assume that values in different patients fail (or not) independently of each other. Let X be the number of patients in a group of 500 who will need another valve replacement within 15 years.

3. You deal 10 cards from a shuffled deck and count the number of red cards.

Answer 1

Conditions of binomial distribution :

1. only one outcome for each trial.
2. each trial has same probability of success.
3. each trial is independent of each other.

a) You toss a balanced coin 50 times and count the number of heads.

Since, the number of heads are drawn in toss are independent of each other and there are only two possible outcomes and one of the two possible in each trial. The probability of drawing head in each trial is same as 0.5. Therefore, the event is binomial with n = 50 and p = 0.5

b) Replacement heart valves have a probability of 0.77 of performing well for 15 years. It is reasonable to assume that values in different patients fail (or not) independently of each other. Let X be the number of patients in a group of 500 who will need another valve replacement within 15 years.

Since, the number of different patients fail ( or not ) are independent of each other and there are only two possible outcomes ( fails or not ) and one of the two possible in each trial. The probability of patient well for 15 years in each trial is same as 0.77 Therefore, the event is binomial with n = 500 and p = 0.77

c) You deal 10 cards from a shuffled deck and count the number of red cards.

Since, the number of red cards drawn are not independent of each other although there are only two possible outcomes ( red or not ) and one of the two possible in each trial. But the probability of drawing red card in each trial changes with every draw. Therefore, the event is not binomial.