You roll two fair dice, a green one and a red one. (a) What...

You roll two fair dice, a green one and a red one.
(a) What is the probability of getting a sum of $7 ?$
(b) What is the probability of getting a sum of $11 ?$
(c) What is the probability of getting a sum of 7 or $11 ?$ Are these outcomes mutually exclusive?

Key Concepts

Addition Rule for Probabilities

The addition rule for probabilities stipulates that for mutually exclusive events, the probability that one or another will occur is the sum of the probabilities of each individual event. This rule simplifies the calculation when events do not overlap in outcomes.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. In the dice roll context, if the outcome is a sum of 7, it cannot simultaneously be a sum of 11, making the events mutually exclusive. This concept is used to decide whether to add the probabilities directly when calculating the probability of either event occurring.

Sample Space

The sample space in probability theory is the set of all possible outcomes from an experiment. In the context of rolling two dice, each die's result combines to form ordered pairs which represent every potential outcome, laying the foundation for calculating probabilities of specific events.

Favorable Outcomes

Favorable outcomes are those outcomes within the sample space that satisfy the event or condition of interest. When calculating probabilities for sums of the dice rolls, one identifies the set of ordered pairs that sum to the target number, and this set is used in determining the probability.

Probability Calculation

Probability calculation involves determining the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This ratio gives the likelihood of the occurrence of a specific event, such as obtaining a sum of 7 or 11 when rolling dice.

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