1. A customer comes into Pierre's and orders a random assortment of 6 danish. At the time she comes in, there are 26 danish sitting out: 12 raspberry, 8 cheese, and 6 cinnamon. Assume the danish are not replaced.



a) What is the probability that the third and fifth danish selected for the customer's random assortment will be raspberry and the other four will not be raspberry?



b) What is the probability that the random assortment will contain at least 1 cheese danish?





2. At Pierre's Coffee Shop, 57% of customers order coffee, 24% order tea and 16% order orange juice, while 3% do not order anything to drink. In addition, 43% order a muffin, 34% order a danish and 14% order a bagel, while 9% do not order anything to eat. Finally, 26% of customers order both coffee and a danish, and 47% of people order a bagel given that they have ordered tea. Assume that an individual customer will not order more than once drink or more than one breakfast roll.



a) When the coffee shop opens, what is the probability that the first person to not order anything to eat is the fifth person to enter the shop?



b) What is the probability that a particular customer will order coffee given that he did NOT order a danish?

I will do #1

#1
let R represent picking a raspberry, N a non-raspberry
12R , 14N

a) so you want: NNRNRN
prob = (14/26)(13/25)(12/24)(12/23)(11/22)(11/21)
= .01913

Prob(none cheese)= (18*17*16*15*14*13)/(26*25*24*23*22*21)
= .08063
at least one cheese
= 1 - .08063
= .91936

a) To find the probability that the third and fifth danish selected for the customer's random assortment will be raspberry and the other four will not be raspberry, we need to calculate the probability of each specific outcome and then add them together.

First, let's calculate the probability of selecting a raspberry danish on the third and fifth attempts. Since there are 12 raspberry danish initially and no replacement occurs, the probability of selecting a raspberry danish on the third attempt is 12/26. After the third selection, there are 25 remaining danish, including 11 raspberry ones. So the probability of selecting a raspberry danish on the fifth attempt is 11/25.

Next, we need to calculate the probability of not selecting a raspberry danish for the other four danish. After the first selection, there are 25 remaining danish, including 11 raspberry ones. So the probability of not choosing a raspberry danish on the first attempt is 14/25. After the second selection, there are 24 remaining danish, including 11 raspberry ones. So the probability of not choosing a raspberry danish on the second attempt is 13/24. Similarly, for the fourth and sixth attempts, we have 23 remaining danish after the third selection and 22 remaining danish after the fifth selection, respectively. The probabilities of not choosing a raspberry danish on the fourth and sixth attempts are 12/23 and 11/22, respectively.

To calculate the overall probability, we multiply all these probabilities together:

P(raspberry, not raspberry, raspberry, not raspberry, raspberry, not raspberry) = (12/26) * (14/25) * (13/24) * (12/23) * (11/25) * (11/22) ≈ 0.0522

So the probability that the third and fifth danish selected for the customer's random assortment will be raspberry and the other four will not be raspberry is approximately 0.0522.

b) To find the probability that the random assortment will contain at least 1 cheese danish, we need to calculate the probability of the complementary event, i.e., the probability that the random assortment contains no cheese danish, and then subtract it from 1.

Since there are 8 cheese danish initially, the probability of not selecting a cheese danish on the first attempt is 18/26. After the first selection, we have 25 remaining danish, including 8 cheese ones, so the probability of not selecting a cheese danish on the second attempt is 17/25. Similarly, for the third, fourth, fifth, and sixth attempts, the probabilities of not selecting a cheese danish are 16/24, 15/23, 14/22, and 13/21, respectively.

To calculate the overall probability of not selecting a cheese danish in any of the six attempts, we multiply these probabilities together:

P(no cheese danish) = (18/26) * (17/25) * (16/24) * (15/23) * (14/22) * (13/21) ≈ 0.1211

Therefore, the probability of the random assortment containing at least 1 cheese danish is:

P(at least 1 cheese danish) = 1 - P(no cheese danish) = 1 - 0.1211 ≈ 0.8789

So the probability that the random assortment will contain at least 1 cheese danish is approximately 0.8789.

------------

a) To determine the probability that the first person to not order anything to eat is the fifth person to enter the shop, we need to multiply the probabilities of the first four people ordering something to eat (not nothing), and then multiply it by the probability of the fifth person ordering nothing to eat.

Given that 43% of people order a muffin, 34% order a danish, and 14% order a bagel, it means that 100% - (43% + 34% + 14%) = 9% of people order nothing to eat.

Therefore, the probability of the first person to order something to eat is 1 - 9% = 91%.

Since each person's order does not depend on the previous person's order, we can assume independence. Thus, the probability of the first four people ordering something to eat is:

P(first four people ordering something to eat) = (91%)^4 ≈ 0.6832

Finally, the probability that the fifth person orders nothing to eat is 9%.

Therefore, the probability that the first person to not order anything to eat is the fifth person to enter the shop is:

P(fifth person not ordering anything to eat) = P(first four people ordering something to eat) * P(fifth person ordering nothing to eat) ≈ 0.6832 * 9% ≈ 0.0615

So the probability that the first person to not order anything to eat is the fifth person to enter the shop is approximately 0.0615.

b) To determine the probability that a particular customer will order coffee given that he did NOT order a danish, we need to calculate the conditional probability.

First, let's determine the probability of not ordering a danish, which we can calculate as 1 - 34% (the percentage of customers who order a danish) = 66%.

Given that 57% of customers order coffee, we can assume the probability of ordering coffee is 57%.

Now, we can use the conditional probability formula:

P(coffee | not ordering a danish) = P(coffee and not ordering a danish) / P(not ordering a danish)

Since ordering a danish and ordering coffee are assumed to be independent events, we can simply multiply the probabilities together:

P(coffee and not ordering a danish) = P(coffee) * P(not ordering a danish) = 57% * 66% = 37.62%

Therefore, the probability that a particular customer will order coffee given that he did NOT order a danish is approximately 37.62%.