In a bag of peanut M & M's, there are 80 M & Ms, with 11 red ones, 12 orange ones, 20 blue ones,

step 1 We have a bag with 80 M &Ms, and 11 are red, 12 are orange, 20 are blue, 11 are green, 18 are ye

In a bag of peanut M & M's, there are 80 M & Ms, with 11 red...

In a bag of peanut $M$ \& M's, there are $80 \mathrm{M} \& \mathrm{Ms}$, with 11 red ones, 12 orange ones, 20 blue ones, 11 green ones, 18 yellow ones, and 8 brown ones. They are mixed up so that each candy piece is equally likely to be selected if we pick one. (a) If we select one at random, what is the probability that it is red? (b) If we select one at random, what is the probability that it is not blue? (c) If we select one at random, what is the probability that it is red or orange? (d) If we select one at random, then put it back, mix them up well (so the selections are independent) and select another one, what is the probability that both the first and second ones are blue? (e) If we select one, keep it, and then select a second one, what is the probability that the first one is red and the second one is green?

In a bag of peanut $M$ \& M's, there are $80 \mathrm{M} \& \mathrm{Ms}$, with 11 red ones, 12 orange ones, 20 blue ones, 11 green ones, 18 yellow ones, and 8 brown ones. They are mixed up so that each candy piece is equally likely to be selected if we pick one.
(a) If we select one at random, what is the probability that it is red?
(b) If we select one at random, what is the probability that it is not blue?
(c) If we select one at random, what is the probability that it is red or orange?
(d) If we select one at random, then put it back, mix them up well (so the selections are independent) and select another one, what is the probability that both the first and second ones are blue?
(e) If we select one, keep it, and then select a second one, what is the probability that the first one is red and the second one is green?

Key Concepts

Sample Space

The sample space is the collection of all possible outcomes of a random experiment. In probability problems, it is essential to define the total number of outcomes, as probabilities are computed by comparing the favorable outcomes to the total number of outcomes.

Favorable Outcomes

Favorable outcomes are the specific outcomes of interest for an event. When calculating probability, counting the number of favorable cases in the sample space enables the determination of the likelihood that the event occurs.

Probability as a Ratio

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This fundamental concept allows one to compute the chance of various events occurring in any random experiment.

Complementary Probability

Complementary probability refers to the concept that the probability of an event not occurring is equal to one minus the probability that the event does occur. This rule is particularly useful when it is easier to calculate the probability of the event rather than its complement.

Addition Rule for Mutually Exclusive Events

The addition rule for mutually exclusive events states that if two events cannot happen at the same time, the probability that one or another occurs is the sum of their individual probabilities. This rule is used when considering events like selecting one candy of a specific color or another, which cannot both happen simultaneously in one draw.

Multiplication Rule for Independent Events

When two events are independent—that is, the outcome of one does not affect the outcome of the other—the multiplication rule states that the probability of both events occurring is the product of their individual probabilities. This rule is used in experiments with replacement, where each draw is an independent event.

Conditional Probability in Dependent Events

Conditional probability comes into play when events are dependent on one another, such as when selections are made without replacement. In these cases, the probability of the second event is affected by the outcome of the first event, and calculations must account for the reduced sample space after the first selection.

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