Chapter 0.6, Problem 2E

(a)

To determine

Probability for Yana will first select a blue sock, then a black sock, and then another blue sock.

Answer to Problem 2E

  $\frac{5}{204}$

Explanation of Solution

Given information:

Number of black socks are 4.

Number of blue socks are 6.

Number of white socks are 8.

3 socks are selected at random without replacement.

Calculations:

Since the socks are being selected without replacement, the events are dependent.

Total number of socks is 18.

Now,

Probability for taking out blue sock,

  $P\left(blue\right)=\frac{6}{18}=\frac{1}{3}$

Probability for taking out black sock after blue sock,

  $P\left(black|blue\right)=\frac{4}{17}$

Probability for taking out blue sock again after one blue and one black sock,

  $P\left(blue|\left(blackandblue\right)\right)=\frac{5}{16}$

In the last step, we have 5 over 16.

At the beginning,

We had 6 blue socks.

After that

We made first pick and took out blue sock.

Thus,

The total number of socks decrease by 1 and number of blue socks also decrease by 1.

There were 18 socks in the beginning.

After first pick, we were left with 17 socks.

Now,

Calculate the probability without replacement by multiplying the above probabilities:

  $P\left(withoutreplacement\right)=\frac{1}{3}\cdot \frac{4}{17}\cdot \frac{5}{16}=\frac{5}{204}$

Thus,

The probability without replacement is $\frac{5}{204}$ .