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Chapter 4: Problem 5

A jar contains four coins: a nickel, a dime, a quarter, and a half-dollar. Three coins are randomly selected from the jar. a. List the simple events in \(S\). b. What is the probability that the selection will contain the half-dollar? c. What is the probability that the total amount drawn will equal \(60 \phi\) or more?

Short Answer

Expert verified
Answer: The probability of selecting a half-dollar coin is 3/4, and the probability of having a total amount of 60 cents or more is also 3/4.

Step by step solution

01

List the simple events in the \(S\)

To find all possible simple events, we will list all the possible combinations of selecting three coins from the jar. The simple events can be represented as ordered triples, \((N, D, Q, H)\) where \(N, D, Q, H\) stand for a nickel, dime, quarter, and half-dollar respectively. The sample space \(S\) contains all outcomes of selecting three coins: \(S=\{(N,D,Q), (N,D,H), (N,Q,H), (D,Q,H)\}\)
02

Find the probability of the half-dollar being in the selection

To find the probability of the half-dollar being in the selection, we will simply count the number of simple events that include the half-dollar and divide by the total number of simple events in \(S\). Out of the four simple events, three of them have the half-dollar in their selection, so the probability is: \(P(H) = \frac{3}{4}\)
03

Calculate the total amount for each simple event

To find the probability of having a total amount of \(60\phi\) cents or more, we will first need to calculate the total amount for each simple event: 1. \((N,D,Q)\): The total sum is \(5 + 10 + 25 = 40\phi\) cents. 2. \((N,D,H)\): The total sum is \(5 + 10 + 50 = 65\phi\) cents. 3. \((N,Q,H)\): The total sum is \(5 + 25 + 50 = 80\phi\) cents. 4. \((D,Q,H)\): The total sum is \(10 + 25 + 50 = 85\phi\) cents.
04

Calculate the probability of having a total amount of \(60\phi\) cents or more

Now we can find the probability of having a total amount of \(60\phi\) cents or more by counting the number of simple events that meet this condition and dividing by the total number of simple events: There are 3 out of 4 simple events that meet this condition, so the probability is: \(P(60\phi\text{ cents or more}) = \frac{3}{4}\)

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