Question

Which of the following situations describes a random variable that has a binomial distribution?

a)A fair coin is tossed 10 times. The variable X is the number of heads resulting from these 10 tosses.

b)A couple will keep having children until they have three girls or five children. The variable X is the number of children in the family.

c)The variable X is the number of clients in the bank between 10:00 a.m. and 11:00 a.m.

d)Fifteen cards are taken from the deck without placing them back in the deck. The variable X is the number of ace outcomes.

Answer 1

Binomial distribution need to satisfy these 4 conditions

1. Fixed number of trials
2. Independent trials
3. Two different classifications
4. The probability of success stays the same for all trials

a)A fair coin is tossed 10 times. The variable X is the number of heads resulting from these 10 tosses.

fixed number of trails = 10

indepenent trials because last toss doesn't affect upcomming tosses

two different classifications; head/tail

probability of success stays the same for all trial since coin is not changed

Hence this is a binomial distribution

b)A couple will keep having children until they have three girls or five children. The variable X is the number of children in the family.

there is no fixed number of trial in this situation

So this is not a binomial distribution

c)The variable X is the number of clients in the bank between 10:00 a.m. and 11:00 a.m.

there is no fixed number of trial in this situation

So this is not a binomial distribution

d) Fifteen cards are taken from the deck without placing them back in the deck. The variable X is the number of ace outcomes.

fixed number of trails = 15

non independent trials since the card is not replaced after choosing

two different classifications; ace/not ace

probability of success doesn't stays the same for all trial since cards are not replaced

So this is not a binomial distribution