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Computing the Probability of the Union of Two Events · Physics Playground

Exercises Sequences, counting and probability

Example 3 from Algebra and Trigonometry, 13.7 Probability

Computing the Probability of the Union of Two Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is $\frac{1}{4} .$ There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is $\frac{1}{13} .$

The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7 is $\frac{1}{52} .$ Substitute $P (H) = \frac{1}{4}, P (7) = \frac{1}{13}, and P (H \cap 7) = \frac{1}{52}$ into the formula.

$\begin{array}{l} P (E \cup^{} F) = P (E) + P (F) - P (E \cap^{} F) \\ = \frac{1}{4} + \frac{1}{13} - \frac{1}{52} \\ = \frac{4}{13} \end{array}$

The probability of drawing a heart or a 7 is $\frac{4}{13} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

How did it go?

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From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

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