Chapter 13.10, Problem 21PPS

To determine

The probability of the socks which will randomly pick a pair of socks that match.

Answer to Problem 21PPS

The probability of probability of the socks which will randomly pick a pair of socks that match isP (pair of socks) = $\frac{22}{69}$

Explanation of Solution

Given:

Blue socks: $6$

Black socks: $8$

White socks: $10$

Total Marbles: $6+8+10=24$

Calculation:

Fraction Form:

Calculate the probability of getting the pair of the socks as it is given that there total $24$ socks and whether blue, black or white to get a pair, pick $2$ simultaneously.

P (Pair of Blue socks),

For this take the number of total of blue socks and total number of socks.

P (First sock is blue) = $\frac{6}{24}=\frac{1}{4}$

Now remove the $1$ sock from blue sock, the probability is,

P (Second sock is blue) = $\frac{5}{23}$

P (Pair of Blue Socks) = P (First) * P (Second)

  $\begin{array}{l}=\frac{1}{4}\times \frac{5}{23}\\ =\left(\frac{1.5}{4.23}\right)\\ =\frac{5}{92}\end{array}$

So the P (Pair of Blue socks) $=\frac{5}{92}$

P (Pair of Black socks)

For this take the number of total of black socks and total number of socks.

P (First sock is black) = $\frac{8}{24}=\frac{1}{3}$

Now remove the $1$ sock from black sock, the probability is,

P (Second sock is black) = $\frac{7}{23}$

P (Pair of Black Socks) = P (First) * P (Second)

  $\begin{array}{l}=\frac{1}{3}\times \frac{7}{23}\\ =\left(\frac{1.7}{3.23}\right)\\ =\frac{7}{69}\end{array}$

So the P (Pair of Black socks) $=\frac{7}{69}$

P (Pair of White socks)

For this take the number of total of white socks and total number of socks.

P (First sock is white) = $\frac{10}{24}=\frac{5}{12}$

Now remove the $1$ sock from white sock, the probability is,

P (Second sock is black) = $\frac{9}{23}$

P (Pair of white Socks) = P (First) * P (Second)

  $\begin{array}{l}=\frac{1}{4}\times \frac{5}{23}\\ =\left(\frac{5.9}{12.23}\right)\\ =\frac{45}{276}=\frac{15}{92}\end{array}$

So the P (Pair of White socks) $=\frac{15}{92}$

Now, as having the probability of the all colour socks, these $3$ events at the same time, this is an example of the mutually exclusive event. So with the based on these events the probability of a picking any pair of socks is given as,

P (Pair of Socks):

P (Pair of Socks) = P (Blue) + P (Black) + White

  $\begin{array}{l}=\frac{5}{92}+\frac{7}{69}+\frac{15}{92}\\ =\left(\frac{5+15}{92}\right)+\frac{7}{69}\\ =\frac{20}{92}+\frac{7}{69}\\ =\frac{5}{23}+\frac{7}{69}\\ =\frac{5\times 3}{23\times 3}+\frac{7}{69}\\ =\frac{15}{69}+\frac{7}{69}\\ =\left(\frac{15+7}{69}\right)\\ =\frac{22}{69}\end{array}$

Conclusion:

Hence, P (pair of socks)= $\frac{22}{69}$