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}.