Two coins are to be flipped. The first coin will land on...

Two coins are to be flipped. The first coin will land on heads with probability $.6,$ the second with probability $.7 .$ Assume that the results of the flips are independent, and let $X$ equal the total number of heads that result.
(a) Find $P\{X=1\}$
(b) Determine $E[X]$

Key Concepts

Independence of Events

This concept refers to the property that the outcome of one event does not affect the outcome of another. In the context of coin flips, it means the result of one coin does not influence the result of the other, allowing for the probabilities of combined outcomes to be computed as the product of their individual probabilities.

Bernoulli Random Variables

A Bernoulli random variable is used to model experiments that have two outcomes, often labeled as 'success' and 'failure'. In coin flips, each flip is a Bernoulli trial with a specific probability of landing heads (success), and this framework is essential for setting up and analyzing problems that involve binary outcomes.

Probability of Compound Events

This concept involves determining the probability of an event that can occur in multiple distinct ways. For example, calculating the probability of getting exactly one head requires considering the mutually exclusive events of one coin landing heads and the other tails, and then summing the probabilities of these individual events.

Linearity of Expectation

Linearity of expectation is the principle that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether the variables are independent. This property simplifies the computation of the expected number of heads by allowing the addition of the expectations of each individual coin flip.

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