Elizabeth lives in San Francisco and works in Mountain View. In the morning, she has $3$ transportation options (take a bus, a cab, or a train) to work, and in the evening she has the same $3$ choices for her trip home. $\newline$ If Elizabeth randomly chooses her ride in the morning and in the evening, what is the probability that she'll use a cab exactly one time?

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Algebra 2

Find probabilities using the addition rule

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Q. Elizabeth lives in San Francisco and works in Mountain View. In the morning, she has

3

transportation options (take a bus, a cab, or a train) to work, and in the evening she has the same

3

choices for her trip home.

\newline

If Elizabeth randomly chooses her ride in the morning and in the evening, what is the probability that she'll use a cab exactly one time?

Options Combinations: Elizabeth has $3$ options in the morning and $3$ options in the evening, making a total of $3 \times 3 = 9$ possible combinations for her round trip.
Morning and Evening Cabs: To use a cab exactly once, Elizabeth can either take a cab in the morning and not in the evening, or not take a cab in the morning and take it in the evening.
Probability Calculation: The probability of taking a cab in the morning and not in the evening is $\frac{1}{3}$ (for the cab in the morning) * $\frac{2}{3}$ (for not taking the cab in the evening).
Total Probability Calculation: The probability of not taking a cab in the morning and taking it in the evening is also $\frac{2}{3}$ (for not taking the cab in the morning) * $\frac{1}{3}$ (for the cab in the evening).
Total Probability Calculation: The probability of not taking a cab in the morning and taking it in the evening is also $\frac{2}{3}$ (for not taking the cab in the morning) * $\frac{1}{3}$ (for the cab in the evening).Adding these two probabilities together gives us the total probability of using a cab exactly once: $(\frac{1}{3} * \frac{2}{3}) + (\frac{2}{3} * \frac{1}{3})$ .
Total Probability Calculation: The probability of not taking a cab in the morning and taking it in the evening is also $\frac{2}{3}$ (for not taking the cab in the morning) * $\frac{1}{3}$ (for the cab in the evening).Adding these two probabilities together gives us the total probability of using a cab exactly once: $(\frac{1}{3} * \frac{2}{3}) + (\frac{2}{3} * \frac{1}{3})$ .Calculating this, we get $(\frac{2}{9}) + (\frac{2}{9}) = \frac{4}{9}$ .