Probability

(31) A bag contains 8 blue marbles and 7 green marbles. If you draw two marbles without replacement, what is the probability that both marbles are green?

• (a) \(\frac{14}{30}\)
• (b) \(\frac{7}{15}\)
• (c) \(\frac{49}{225}\)
• (d) \(\frac{21}{64}\)

Answer

Correct Answer: (c) \(\frac{49}{225}\)
P(\text{both green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} \times \frac{\text{Number of green marbles - 1}}{\text{Total number of marbles - 1}} = \frac{7}{15} \times \frac{6}{14} = \frac{49}{225}.

(32) In a group of 50 students, 30 like ice cream, and 20 like cake. If 10 students like both ice cream and cake, what is the probability of selecting a student at random who likes either ice cream or cake?

• (a) \(\frac{40}{50}\)
• (b) \(\frac{40}{50}\)
• (c) \(\frac{30}{50}\)
• (d) \(\frac{50}{50}\)

Answer

Correct Answer: (b) \(\frac{40}{50}\)
P(\text{ice cream or cake}) = P(\text{ice cream}) + P(\text{cake}) - P(\text{both}) = \frac{30}{50} + \frac{20}{50} - \frac{10}{50} = \frac{40}{50}.

(33) You have a bag containing 5 red balls, 4 green balls, and 3 blue balls. If you draw three balls without replacement, what is the probability that all three balls are red?

• (a) \(\frac{15}{60}\)
• (b) \(\frac{10}{66}\)
• (c) \(\frac{20}{55}\)
• (d) \(\frac{25}{75}\)

Answer

Correct Answer: (b) \(\frac{10}{66}\)
P(\text{all three red}) = \frac{\text{Number of red balls}}{\text{Total number of balls}} \times \frac{\text{Number of red balls - 1}}{\text{Total number of balls - 1}} \times \frac{\text{Number of red balls - 2}}{\text{Total number of balls - 2}} = \frac{5}{12} \times \frac{4}{11} \times \frac{3}{10} = \frac{10}{66}.

(34) A coin is tossed three times. What is the probability of getting exactly two heads?

• (a) \(\frac{1}{8}\)
• (b) \(\frac{3}{8}\)
• (c) \(\frac{1}{4}\)
• (d) \(\frac{1}{2}\)

Answer

Correct Answer: (b) \(\frac{3}{8}\)
P(\text{exactly two heads}) = P(\text{HHH}) + P(\text{HHT}) + P(\text{HTH}) = \frac{1}{8} + \frac{1}{4} + \frac{1}{4} = \frac{3}{8}.

(35) You draw a card from a standard deck of 52 cards. What is the probability of drawing a face card (jack, queen, or king) or a heart?

• (a) \(\frac{12}{52}\)
• (b) \(\frac{23}{52}\)
• (c) \(\frac{9}{52}\)
• (d) \(\frac{7}{52}\)

Answer

Correct Answer: (b) \(\frac{23}{52}\)
P(\text{face card or heart}) = P(\text{face card}) + P(\text{heart}) - P(\text{both}) = \frac{12}{52} + \frac{13}{52} - \frac{3}{52} = \frac{23}{52}.