You decide to start taking coins out of your piggy bank.

Question

You decide to start taking coins out of your piggy bank. Currently, your piggy bank has 47 coins, of which 13 are toonies and 7 are loonies. Suppose that when you take a coin out, you do not put it back in the piggy bank; you are picking coins without replacement. (Round your answers to 4 decimal places, if needed.) (a) What is the probability that the third coin you pick is the first toonie you find? (b) What is the probability that at least one of the first two coins you pick is a toonie? (c) What is the probability that the first three coins you pick are all either loonies or toonies?

Answer 1

Hey! This looks like a fun/challenging question.

There are several important details to pull from this question:

We are selecting the coins without replacement. This means that the total number of coins will change as we draw our coins.

We will use our probability rules to find a solution.

The first rule is the Compliment Rule, which shows us that if the probability of rolling a 1 with a six-sided die is 1/6, then the probability of not rolling a 1 with a six-sided die is 5/6.

Let’s play with the numbers: Coins can be toonies, loonies, or not_toonies_or_loonies.

P(toonie) = 13/47

P(loonie) = 7/47

P(not_toonie_or_loonie) = 27/40

(a) What is the probability that the third coin you pick is the first toonie you find?

First draw – we want the probability of not drawing a toonie, which is the compliment of drawing a toonie: (if P(toonie) = 13/47, then P(not toonie) = 34/47)

First draw: P(not toonie) = 34/47

Second draw – we want the probability of not drawing a toonie again, which is the compliment of drawing a toonie (remember that we are not replacing coins as we draw them which impacts our numerator and our denominator):

Second draw: P(not toonie) = 33/46

Third draw – now, we want the probability of drawing a toonie, on this draw (remember that we are not replacing coins as we draw them which impacts our numerator and our denominator):

Third draw: P(toonie) = 13/45

To find the solution multiple the probability of each draw!

(b) What is the probability that at least one of the first two coins you pick is a toonie?

For this part, let’s think about the sample space, the possible outcomes for our scenario. We want to find the outcomes with at least on toonie. Remember, we are drawing coins without replacement – so be careful!

I like to use a grid or decision tree to identify the sample space. Because we’re only drawing twice, I chose to use a grid to find all the possible outcomes:

T L X

T (T,T) (T,L) (T,X)

L (L,T) (L,L) (L,X)

X (X,T) (X,L) (X,X)

Next, we can calculate the probability of each outcome in the sample space:

Example of finding the probabilities:

P(T,T)=(13/47)*(12/46);

P(L,T) =(7/47)*(13/46)

P(X,X) =(27/47)*(26/46)

P(T,T) 0.07215541

P(L,T) 0.04209066

P(X,T) 0.16234968

P(T,L) 0.04209066

P(L,L) 0.01942646

P(X,L) 0.08741906

P(T,X) 0.16234968

P(L,X) 0.08741906

P(X,X) 0.32469935

To check your work, the sum of the probabilities within the sample space should be 1.

Finally, we add up the probabilities for the outcomes in which we have at least one toonie in the first two draws.

(c) What is the probability that the first three coins you pick are all either loonies or toonies?

P(T,T,T) or P(L,L,L)

P(T,T,T) = (13/47) * (12/46) * (11/45) =

P(L,L,L) = (7/47) * (6/46) * (5/45) =

Add the two probabilities together to find the probability of either all loonies or toonies!