In Exercises 5–36, express all probabilities as fractions.

Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

Six peas are to be chosen out of eight.

Out of the eight peas, four are smooth yellow, and four are wrinkled green.

When r units are to be chosen from a set of n units without replacement, then the combination rule is used to find the total number of ways in which the selections can be made.

The formula is shown below:

${}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Here,the order of the selections is not considered.

Let A be the event of selecting all peas of the same type. Two cases are possible:

Case 1: Selecting all smooth yellow peas

The total number of peas is eight.

The number of peas to be chosen is four.

The number of ways in which four peas can be chosen (in any order) is

$$\begin{gathered} {}^8{C}_4=\frac{8!}{\left(8-4\right)!4!} \\ =70. \end{gathered}$$

The number of ways in which all smooth yellow peas are selected is one.

The probability of selecting all smooth yellow peas is

$P\left(\text{all}\text{smooth}\text{yellow}\right)=\frac{1}{70}$

Case 2: Selecting all wrinkled green peas

The number of ways in which all wrinkled peas can be chosen is one.

The probability of selecting all wrinkled green peas is

$P\left(\text{all}\text{wrinkled}\text{green}\right)=\frac{1}{70}$

The probability of selecting all peas of the same type is the sum of the probabilities of selecting all smooth yellow peas or all wrinkled green peas. The calculation is shown below:

$$\begin{gathered} P\left(A\right)=\frac{1}{70}+\frac{1}{70} \\ =\frac{1}{35} \\ =0.0286 \end{gathered}$$

Therefore, the probability of selecting all four peas of the same type is 0.0286.