Question

4. (10 pts) For each of the following scenarios, determine whether the binomial distribution is an appropriate model (distribution) for the random variable X. If yes, identify the values of the parameters n and p. A. A fair coin is flipped 10 times. Let X = the number of times the coin comes up tails. B. A fair coin is flipped multiple times. Let X = the number of times the coin needs to be flipped until we see 10 tails. C. A standard roulette wheel with one ball in it is turned six times. Let X = the number of times the ball lands on red.

Answer 1

a) The number of times the coin come up fair tails is a binomial distribution given as:

$X \sim Bin(n = 10, p= 0.5)$

This is because the probability of getting a tails on each trial remains same and is equal to 0.5 for a fair coin. Also each trial is independent of each other. Also the number of trials is constant here equal to 10.

b) The number of times we need to toss to have 10 tails does not have a fixed number of trials, as there could be 10 tosses in which we get 10 tails or even 20 tosses where we get 10 tails. Therefore this is not a binomial distribution here.

c) The probability of landing red remains same for each spin:
P(red) = n(Red) / n(Total) = 18/38 = 9/19

Also as we are rolling it 6 times, the number of trials remains same. Therefore the distribution here is binomial given as:

$X \sim Bin(n = 6, p= 9/19)$