8. (5pts) Each time a shopper purchases a tube of toothpaste, he chooses either

Question

8. (5pts) Each time a shopper purchases a tube of toothpaste, he chooses either brand A or brand B. Suppose that for each purchase after the first, the probability is 1/3 that he will choose the same brand that he chose on his preceding purchase and the probability is 2/3 that he will switch brands. If he is equally likely to choose either brand A or brand B on his first purchase, what is the probability that both his first and second purchases will be one brand (either Brand A or B) and both his third and fourth purchases will be the other band (either Brand B or A)?

Explanation / Answer

P(first two barnd are same and last two are same and other than first two brands)

=P(AABB+BBAA) =P(chooses A with probability 1/2 ; then same brand A with probability 1/3 ; chooses B with probability 2/3 then same brand B with probability 1/3+ same as for choosing starting with B)

=(1/2)*(1/3)*(2/3)*(1/3)+(1/2)*(1/3)*(2/3)*(1/3) =4/54 =2/27

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