Question

A bag contains letter tiles. Forty-four of the tiles are vowels, and $56$ are consonants. Seven tiles are picked at random. You want to know the probability that four of the seven tiles are vowels. What is the group of interest, the size of the group of interest, and the size of the sample?

Short answer

The probability that four of the seven tiles are vowels is$0.2351$

Step-by-step solution

Content Introduction

We are given,

A bag contains letter tiles.

$44$of the tiles are vowels, and $56$are consonants.

$7$ tiles are picked at random.

Content Explanation

Let X the number of vowels in the $7$letter tiles, the number of vowels are the groups of interest and the sample size X takes the values $0,1,2,3,4,5,6,7$

Here, X follows a hypergeometric distribution with $K=44$boys in population, $N=100$.

where $k=4$successes from the sample size $n=7$

The probability mass function of hypergeometric distribution of X is:

$$\begin{gathered} P(X=k)=\frac{\left({}^{k}C_k\right)\left({}^{n-k}C_{n-k}\right)}{{}^{n}C_n} \\ P(X=4)=\frac{\left({}^{44}C_4\right)\left({}^{100-44}C_{7-4}\right)}{{}^{100}C_7} \\ =0.2351 \end{gathered}$$