Chapter 11.4, Problem 18PPS

To determine

What is the probability that $20$ of the flowers will be blue?

Answer to Problem 18PPS

  $\boxed{0.1316or13.16\%}$

Explanation of Solution

Given information:

Peter is planting $24$ irises in his front yard. The flowers he bought were a combination of two varieties, blue and white. The flowers are not blooming yet, but Peter knows that the probability of having a blue flower $75\%$ . What is the probability that $20$ of the flowers will be blue?

Calculation:

Cosider a person P is planting $24$ irises in his garden.

There are $2$ varieties if irises, blue and white among the $24$ irises.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure which is equal to,

  $q=1-p$

Since the probability of having a blue flower is $75\%$ ,so,

  $\begin{array}{l}p=75\%\\ =0.75\end{array}$

The probability of failure is,

  $q=1-p$

  $=1-0.75$

  $=0.25$

Substitute $p=0.75,q=0.25,k=20,n=24$ in the above formula to find the probability that $20$ of the $24$ flowers will be blue.

  $\begin{array}{l}P(20){=}^{24}{C}_{20}{(}^{0.75}{(}^{0.25}\\ =10,626{(}^{0.75}{(}^{0.25}\\ =0.1316\\ =13.16\%\end{array}$

Hence, the probability that $20$ of the $24$ flowers will be blue is $\boxed{0.1316or13.16\%}$