Johnny goes to the bakery every morning and always buys either one donut or one

Question

Johnny goes to the bakery every morning and always buys either one donut or one bagel and nothing else. Whether he buys a donut or a bagel is independent of what he bought on previous days. The owner of the shop has kept track of Johnny's purchases and Johnny purchased a bagel on 40 of the last 100 days. Jane goes to the bakery with Johnny each morning and also buys either a bagel or a donut and nothing else. (Assume Johnny still buys a bagel or donut us described in the previous problem.) Let X = the number of bagels sold to Jane and Johnny on the next 2 days. Is X a random variable? Why or why not? Jane is twice as likely to buy a bagel as she is to buy a donut. What is the probability that Jane buys a bagel?

Explanation / Answer

Question 6

Part A:

Yes X is a random number because there are 4 possibilities of total number of bagels sold to Jane and Johnny on the next two days. The total number of bagels sold can be 1, 2, 3, or 4.

X = 1 when for example,either Johnny or Jane purchases Bagel only on either day 1 or day 2

X = 2 when for example, Johnny purchases bagel on both day 1 and day 2 while Jane purchases Donut on both days

X = 3 when for example, Johnny purchases bagel on both days, while Jane purchases it on either day 1 or day 2

X = 4 when for example, Johnny and Jane purchases Bagel on both the days

Since there is no single value of X that can be expected with 100% surety, hence X is a random variable.

Part B:

Let probability of buying donut by Jane be "p".

Since probability of buying bagel is twice that of buying donut, probabilty of buying bagel is "2p"

Now there are only two possibilities, Jane either buys donut or bagel.

Hence,

Probability of Buying Donut + Probabaility of Buying Bagel = 1

p + 2p = 1

3p = 1

Or, p = 1/3

Probabillity of buying bagel by Jane = 2p = 2*(1/3) = 2/3 = 0.67 (Answer)

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