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You pick a card at random. $\newline$ $678$ $\newline$ What is $P(8$ or less than $8)$ ? $\newline$ Choices: $\newline$ (A) $60\%$ $\newline$ (B) $70\%$ $\newline$ (C) $100\%$ $\newline$ (D) $30\%$

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Probability of mutually exclusive events and overlapping events

Full solution

Q. You pick a card at random.

\newline

678

\newline

What is

P(8

or less than

8)

?

\newline

Choices:

\newline

(A)

60\%

\newline

(B)

70\%

\newline

(C)

100\%

\newline

(D)

30\%

Understand the problem: Step $1$ : Understand the problem. $\newline$ We need to find the probability of picking a card that is an $8$ or less than $8$ from a standard deck of cards.
Calculate favorable outcomes: Step $2$ : Calculate the total number of favorable outcomes. $\newline$ In a standard deck, there are $4$ suits, and each suit has cards numbered from $1$ to $10$ , plus a Jack, Queen, King, and Ace. The Ace is typically considered as $1$ . So, cards less than or equal to $8$ are $1, 2, 3, 4, 5, 6, 7,$ and $8$ . That's $8$ cards per suit. $\newline$ $8$ cards/suit $\times 4$ suits = $1$ $0$ cards.
Calculate total outcomes: Step $3$ : Calculate the total number of possible outcomes. $\newline$ A standard deck has $52$ cards in total.
Calculate probability: Step $4$ : Calculate the probability. $\newline$ Probability ( $P$ ) of picking a card that is $8$ or less = Number of favorable outcomes / Total number of outcomes $\newline$ $P = \frac{32}{52}$ $\newline$ $P = 0.615384615\ldots$
Convert to percentage: Step $5$ : Convert the probability to a percentage. $\newline$ $P \times 100 = 61.54\%$ $\newline$ This rounds to approximately $60\%$ .

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