Probability

(11) You have a standard deck of 52 playing cards. What is the probability of drawing a red card (hearts or diamonds) at random?

• (a) \(\frac{26}{52}\)
• (b) \(\frac{13}{52}\)
• (c) \(\frac{39}{52}\)
• (d) \(\frac{1}{2}\)

Answer

Correct Answer: (a) \(\frac{26}{52}\)
P(\text{red card}) = P(\text{hearts or diamonds}) = \frac{26}{52} = \frac{1}{2}.

(12) You flip a fair coin three times. What is the probability of getting exactly two heads?

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

Answer

Correct Answer: (c) \(\frac{3}{4}\)
P(\text{exactly two heads}) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 3 \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{4}.

(13) You have a bag with 8 red balls, 5 green balls, and 4 blue balls. If you draw one ball at random, what is the probability that it is not green?

• (a) \(\frac{17}{17}\)
• (b) \(\frac{8}{17}\)
• (c) \(\frac{5}{17}\)
• (d) \(\frac{4}{17}\)

Answer

Correct Answer: (a) \(\frac{17}{17}\)
P(\text{not green}) = 1 - P(\text{green}) = 1 - \frac{5}{17} = \frac{17}{17}.

(14) A box contains 5 red balls and 3 blue balls. If you draw two balls with replacement, what is the probability that both balls are red?

• (a) \(\frac{10}{64}\)
• (b) \(\frac{25}{64}\)
• (c) \(\frac{5}{16}\)
• (d) \(\frac{9}{16}\)

Answer

Correct Answer: (b) \(\frac{25}{64}\)
P(\text{first red and second red with replacement}) = P(\text{red}) \cdot P(\text{red}) = \frac{5}{8} \cdot \frac{5}{8} = \frac{25}{64}.

(15) You have a bag with 3 white socks, 4 black socks, and 5 blue socks. If you blindly choose 2 socks, what is the probability that they are both black?

• (a) \(\frac{4}{12}\)
• (b) \(\frac{8}{15}\)
• (c) \(\frac{6}{65}\)
• (d) \(\frac{2}{3}\)

Answer

Correct Answer: (c) \(\frac{6}{65}\)
P(\text{both black}) = \frac{\binom{4}{2}}{\binom{12}{2}} = \frac{6}{65}.