Probability

(36) In a deck of 52 cards, what is the probability of drawing a card that is either a spade or a queen?

• (a) $\frac{4}{52}$
• (b) $\frac{17}{52}$
• (c) $\frac{1}{13}$
• (d) $\frac{9}{52}$

Answer

Correct Answer: (b) $\frac{17}{52}$
$P(\text{spade or queen}) = P(\text{spade}) + P(\text{queen}) - P(\text{both}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{17}{52}.$

(37) You have a bag with 6 red marbles, 5 green marbles, and 4 blue marbles. If you draw one marble at random, what is the probability of getting a red or a blue marble?

• (a) $\frac{5}{15}$
• (b) $\frac{10}{15}$
• (c) $\frac{11}{15}$
• (d) $\frac{9}{15}$

Answer

Correct Answer: (b) $\frac{10}{15}$
$P(\text{red or blue}) = P(\text{red}) + P(\text{blue}) = \frac{6}{15} + \frac{4}{15} = \frac{10}{15}.$

(38) A box contains 3 white balls, 2 black balls, and 4 gray balls. If you randomly select three balls without replacement, what is the probability that at least one ball is black?

• (a) $\frac{2}{6}$
• (b) $\frac{6}{9}$
• (c) $\frac{1}{3}$
• (d) $\frac{7}{9}$

Answer

Correct Answer: (c) $\frac{1}{3}$
To find the probability of at least one ball being black, we can calculate the complementary probability of none of the balls being black and subtract it from 1:
$P(\text{at least one black}) = 1 - P(\text{none of the balls are black})$
$P(\text{none of the balls are black}) = \frac{6}{9} \times \frac{5}{8} \times \frac{4}{7} = \frac{120}{504}$
$P(\text{at least one black}) = 1 - \frac{120}{504} = \frac{384}{504} = \frac{1}{3}.$

(39) A jar contains 8 red balls, 5 blue balls, and 4 green balls. If you draw two balls without replacement, what is the probability that one is red and the other is blue?

• (a) $\frac{40}{85}$
• (b) $\frac{80}{170}$
• (c) $\frac{72}{136}$
• (d) $\frac{14}{30}$

Answer

Correct Answer: (b) $\frac{80}{170}$
$P(\text{one red and one blue}) = P(\text{red and blue}) + P(\text{blue and red})$
$P(\text{red and blue}) = \frac{8}{17} \times \frac{5}{16} = \frac{40}{272}$
$P(\text{blue and red}) = \frac{5}{17} \times \frac{8}{16} = \frac{40}{272}$
$P(\text{one red and one blue}) = \frac{40}{272} + \frac{40}{272} = \frac{80}{272} = \frac{80}{170}.$

(40) You roll a fair six-sided die. What is the probability of rolling a number greater than 4 or an even number?

• (a) $\frac{1}{6}$
• (b) $\frac{4}{6}$
• (c) $\frac{2}{3}$
• (d) $\frac{3}{6}$

Answer

Correct Answer: (b) $\frac{4}{6}$
$P(\text{greater than 4 or even}) = P(\text{greater than 4}) + P(\text{even}) - P(\text{both})$
$P(\text{greater than 4}) = P(\{5, 6\}) = \frac{2}{6}$
$P(\text{even}) = P(\{2, 4, 6\}) = \frac{3}{6}$
$P(\text{both}) = P(\{6\}) = \frac{1}{6}$
$P(\text{greater than 4 or even}) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}.$