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You pick a card at random. $\newline$ $456$ $\newline$ What is $P(\text{even or greater than } 4)$ ? $\newline$ Choices: $\newline$ (A) $90\%$ $\newline$ (B) $100\%$ $\newline$ (C) $70\%$ $\newline$ (D) $130\%$

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Probability of mutually exclusive events and overlapping events

Full solution

Q. You pick a card at random.

\newline

456

\newline

What is

P(\text{even or greater than } 4)

?

\newline

Choices:

\newline

(A)

90\%

\newline

(B)

100\%

\newline

(C)

70\%

\newline

(D)

130\%

Identify total number of cards: Identify the total number of cards and the cards that meet the criteria. $\newline$ In a standard deck, there are $52$ cards. Cards that are even or greater than $4$ include all even cards and any card numbered $5$ or higher in each suit.
Count even cards: Count the even cards. $\newline$ There are $4$ suits, and in each suit, the even-numbered cards are $2$ , $4$ , $6$ , $8$ , $10$ . That's $5$ cards per suit, so $5$ cards $\times$ $4$ suits $2$ $0$ even cards.
Count cards greater than $4$ : Count the cards greater than $4$ . $\newline$ Cards greater than $4$ are $5$ , $6$ , $7$ , $8$ , $9$ , $10$ , Jack, Queen, King in each suit. That's $9$ cards per suit, so $9$ cards $\times$ $4$ suits = $6$ $0$ cards.
Calculate overlap of cards: Calculate the overlap of even cards and cards greater than $4$ . The overlapping cards (even and greater than $4$ ) are $6$ , $8$ , $10$ in each suit. That's $3$ cards per suit, so $3$ cards $\times$ $4$ suits = $12$ cards.
Use inclusion-exclusion principle: Use the inclusion-exclusion principle to find the total number of favorable outcomes. $\newline$ Total favorable = (Number of even cards) + (Number of cards greater than $4$ ) - (Overlap of both) $\newline$ Total favorable = $20 + 36 - 12 = 44$ cards.

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