Consider a bag that contains eight coins: three quarters,...

Consider a bag that contains eight coins: three quarters, two dimes, one nickel, and two pennies. Assume that two coins are picked out of the bag, one at a time, and the first coin is put back into the bag before the second coin is chosen. (a) How many outcomes are there? (Hint: Count the possibilities for the first coin and the possibilities for the second coin.) (b) What is the probability of picking two coins of equal value?

Consider a bag that contains eight coins: three quarters, two dimes, one nickel, and two pennies.
Assume that two coins are picked out of the bag, one at a time, and the first coin is put back into the bag before the second coin is chosen.
(a) How many outcomes are there? (Hint: Count the possibilities for the first coin and the possibilities for the second coin.)
(b) What is the probability of picking two coins of equal value?

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Key Concepts

Classification by Attribute for Counting Favorable Outcomes

When dealing with elements that share common attributes (like coin values), items are grouped based on these attributes to count favorable outcomes. Even if identical items are present, considering their frequency is essential in correctly enumerating the outcomes that satisfy a given condition.

Probability Calculation

Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. This concept underlies the method of determining how likely an event is to occur when assessing all possible outcomes.

Sampling with Replacement

Sampling with replacement means that after an item is chosen in an event, it is returned to the population, keeping the probabilities the same for subsequent draws. This makes the events independent, so the outcome of one draw does not change the probabilities in the next.

Multiplication Principle

This concept involves determining the total number of outcomes for sequential events by multiplying the number of possibilities for each event. When one event does not affect the other (i.e., events are independent), the total outcomes are the product of the outcomes in each step.

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