91Ó°ÊÓ

Each time a shopper purchases a tube of toothpaste, he chooses either brand A or brand B.

He is equally likely to choose either brand A or brand B on his first purchase.

The probability that he will choose the same brand that he chose on his preceding purchase is \(\frac{1}{3}\) .

The probability that he will switch brands is \(\frac{2}{3}\).

Let\({{\bf{A}}_{\bf{i}}}{\bf{,}}{{\bf{B}}_{\bf{i}}}\)be defined as two events such that brand A is selected on i-th turn and brand B is selected on i-th turn.

From the given information,

\(\begin{aligned}{}P\left( {{A_{i + 1}}\left| {{A_i}} \right.} \right) &= P\left( {{B_{i + 1}}\left| {{B_i}} \right.} \right)\\ &= \frac{1}{3}\\P\left( {{A_{i + 1}}\left| {{B_i}} \right.} \right) &= P\left( {{B_{i + 1}}\left| {{A_i}} \right.} \right)\\ &= \frac{2}{3}\end{aligned}\)

Also,

\(\begin{aligned}{}P\left( {{A_1}} \right) &= P\left( {{B_1}} \right)\\ &= \frac{1}{2}\end{aligned}\)

The probability that both his first and second purchases will be brand A and both his third and fourth purchases will be brand B is expressed as:

\(P\left( {{A_1} \cap {A_2} \cap {B_3} \cap {B_4}} \right) = P\left( {{A_1}} \right) \times P\left( {{A_2}\left| {{A_1}} \right.} \right) \times P\left( {{B_3}\left| {{A_1} \cap {A_2}} \right.} \right) \times P\left( {{B_4}\left| {{A_1} \cap {A_2} \cap {B_3}} \right.} \right)\)

From the given information,

\(\begin{aligned}{}P\left( {{A_1} \cap {A_2} \cap {B_3} \cap {B_4}} \right) &= P\left( {{A_1}} \right) \times P\left( {{A_2}\left| {{A_1}} \right.} \right) \times P\left( {{B_3}\left| {{A_1} \cap {A_2}} \right.} \right) \times P\left( {{B_4}\left| {{A_1} \cap {A_2} \cap {B_3}} \right.} \right)\\ &= \frac{1}{2} \times \frac{1}{3} \times \frac{2}{3} \times \frac{1}{3}\\ &= \frac{1}{{27}}\end{aligned}\)

Thus, the required probability is \(\frac{1}{{27}}\) .