Question

Bowl 1 contains 30 vanilla cookies and 10 chocolate cookies.

What is the likelihood of the following: you, without looking, select a vanilla cookie and eat it (i.e. you are selecting without replacement), then select a chocolate cookie, and eat it, and then select a vanilla cookie.

Now, what is the likelihood if instead you had a Bowl 2 which contains 20 of each.

Answer 1

Scenario 1: Bowl 1 : contains 30 Vanilla cookies and 10 Chocolate cookies (total sample - 40)

Suppose Event V is getting a vaniila cookie = P(V) = n(V) / n(S)

Suppose Event C is getting a chocolate cookie = P(C) = n(C)/ n(S)

a) Probability of selecting a vanilla cookie in first selection = P(V1) = 30/40 = 3/4

Since we are not replacing the cookie, Bowl 1 now contains : 29 vanilla cookies and 10 chocolate cookies (Total sample - 39)

The selection of second cookie is Dependent event as the total number to cookies in the jar change & hence the probability will change.

b) Probability of selecting a chocolate cookie in second selection is P(C2 / V1) = 10/39 ( i.e. probability of getting a chocolate cookie considering that first cookie selected is vanilla)

Since we are not replacing the cookie, Bowl 1 now contains : 29 vanilla cookies and 9 chocolate cookies (Total sample - 38)

c) Probability of selecting a vanilla cookie in third selection is P(V3 / V1 C2)= 29/38 (i.e. probability of getting a vanilla cookie considering that the first cookie was vanilla and second was chocolate)

Probability of the scenario 1 (likelihood of - selecting a vanilla cookie, then selecting a chocolate cookie and then selecting a vanilla cookie, all without replacement from bowl 1)

Scenario 2: Bowl 2 : contains 20 Vanilla cookies and 20 Chocolate cookies (total sample - 40)

Suppose Event V is getting a vaniila cookie = P(V) = n(V) / n(S)

Suppose Event C is getting a chocolate cookie = P(C) = n(C)/ n(S)

a) Probability of selecting a vanilla cookie in first selection = P(V1) = 20/40 = 1/2

Since we are not replacing the cookie, Bowl 2 now contains :19 vanilla cookies and 20 chocolate cookies (Total sample - 39)

The selection of second cookie is Dependent event as the total number to cookies in the jar change & hence the probability will change.

b) Probability of selecting a chocolate cookie in second selection is P(C2 / V1) = 20/39 ( i.e. probability of getting a chocolate cookie considering that first cookie selected is vanilla)

Since we are not replacing the cookie, Bowl 2 now contains : 19 vanilla cookies and 19 chocolate cookies (Total sample - 38)

c) Probability of selecting a vanilla cookie in third selection is P(V3 / V1 C2)= 19/38 (i.e. probability of getting a vanilla cookie considering that the first cookie was vanilla and second was chocolate)

Probability of the scenario 2 (likelihood of - selecting a vanilla cookie, then selecting a chocolate cookie and then selecting a vanilla cookie, all without replacement from bowl 2)