Probability

(1) A jar contains 4 blue, 5 green, and 6 red marbles. If a marble is selected at random, what is the probability that it is either green or red?

• (a) \(\frac{5}{15}\)
• (b) \(\frac{11}{15}\)
• (c) \(\frac{4}{15}\)
• (d) \(\frac{9}{15}\)

Answer

Correct Answer: (b) \(\frac{11}{15}\)
P(\text{green or red}) = P(\text{green}) + P(\text{red}) = \frac{5}{15} + \frac{6}{15} = \frac{11}{15}.

(2) A six-sided die is rolled once. What is the probability that the number rolled is an even number greater than 3?

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

Answer

Correct Answer: (b) \(\frac{1}{3}\)
P(\text{even number} > 3) = P(\{4, 6\}) = \frac{2}{6} = \frac{1}{3}.

(3) In a class of 30 students, 18 play basketball, and 16 play football. If 12 students play both sports, what is the probability of selecting a student at random who plays either basketball or football?

• (a) \(\frac{22}{30}\)
• (b) \(\frac{22}{30}\)
• (c) \(\frac{18}{30}\)
• (d) \(\frac{24}{30}\)

Answer

Correct Answer: (b) \(\frac{22}{30}\)
P(\text{basketball or football}) = P(\text{basketball}) + P(\text{football}) - P(\text{both}) = \frac{18}{30} + \frac{16}{30} - \frac{12}{30} = \frac{22}{30}.

(4) Two cards are drawn at random from a deck of 52 cards without replacement. What is the probability of drawing an ace followed by a king?

• (a) \(\frac{1}{132}\)
• (b) \(\frac{4}{663}\)
• (c) \(\frac{1}{221}\)
• (d) \(\frac{1}{26}\)

Answer

Correct Answer: (b) \(\frac{4}{663}\)
P(\text{ace followed by king}) = P(\text{ace first}) \times P(\text{king second} | \text{ace first}) = \frac{4}{52} \times \frac{4}{51} = \frac{4}{663}.

(5) A bag contains 3 red, 2 blue, and 5 green balls. Two balls are drawn randomly. What is the probability that they are not of the same color?

• (a) \(\frac{21}{40}\)
• (b) \(\frac{19}{40}\)
• (c) \(\frac{1}{2}\)
• (d) \(\frac{20}{40}\)

Answer

Correct Answer: (b) \(\frac{19}{40}\)
P(\text{different color}) = 1 - P(\text{same color}) = 1 - \left( \frac{\binom{3}{2} + \binom{2}{2} + \binom{5}{2}}{\binom{10}{2}} \right) = \frac{19}{40}.