Question

You are playing a game called Tsunami Duck Pond. In Tsunami Duck Pond there are 100 ducks that get pummeled by tidal waves. You have to reach your hand into the tsunami and select a duck. If there is a star on the bottom of the duck, you win. The game claims to have 20 ducks with stars among the 100 ducks. After each round you must place the duck back in the tumultuous water. Let W=the number of times you win if you play this game 10 times.
a. Explain why Wis a binomial random variable.
b. Find the probability that you win exactly 3 times.

Answer 1

Answer:

a) W is a binomial random variable because in each trial there are only two possible outcomes, and the probability of a success is always the same.

b) 0.2013 = 20.13% probability that you win exactly 3 times.

Step-by-step explanation:

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

a. Explain why W is a binomial random variable.

For each time a duck is selected, there are only two possible outcomes. Either it has a star, or it does not. Also, after a duck is selected, it is placed back in the water which means that in each trial, the probability of finding a duck with star is the same. This is why W is a binomial random variable.

b. Find the probability that you win exactly 3 times.

Let W=the number of times you win if you play this game 10 times.

This means that $n = 10$

The game claims to have 20 ducks with stars among the 100 ducks.

This means that $p = \frac{20}{100} = 0.2$

We have to find P(X = 3). So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 3) = C_{10,3}.(0.2)^{3}.(0.8)^{7} = 0.2013$

0.2013 = 20.13% probability that you win exactly 3 times.