Q: A student rolls two six-sided dice. What is the probability that the sum of the dice is 7?

$$P(\text{sum}=7) = \frac{6}{36} = \frac{1}{6}$$

Q: A deck of 52 cards is shuffled. What is the probability of drawing either a heart or a king?

$$P(\text{heart or king}) = \frac{13 + 4 - 1}{52} = \frac{16}{52} = \frac{4}{13}$$

Q: A bag contains 3 red, 2 blue, and 5 green marbles. If one marble is drawn at random, what is the probability it is not green?

$$P(\text{not green}) = \frac{3+2}{10} = \frac{5}{10} = 0.5$$

Q: A fair coin is tossed three times. What is the probability of getting exactly two heads?

$$P(\text{exactly 2 heads}) = \binom{3}{2} \left(\frac{1}{2}\$$right)^2$$ \left(\frac{1}{2}\$$right)^1$$ = 3 \times \frac{1}{4} \times \frac{1}{2} = \frac{3}{8}$$

Q: A discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

$$E(X) = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 0.2 + 1.0 + 0.9 = 2.1$$

Q: A binomial experiment consists of 8 trials with probability of success $$p=0.3. $$What is the probability of exactly 2 successes?

$$P(X=2) = \binom{8}{2} (0.3)^2$ (0.7)^6$ = 28 \times 0.09 \times 0.117649 = 0.296$$

Q: A normal distribution has $$\mu=100$$ and $$\sigma=15. $$What is the probability that a randomly selected value is less than $$115$$?

$$z = \frac{115-100}{15} = 1$$; \text{Area to the left of } z=1 \approx 0.8413$

Q: A student scores 80 on a test where the mean is 70 and the standard deviation is 5. What percentile is the student in?

$$z = \frac{80-70}{5} = 2$$; \text{Percentile} \approx 97.72\%$

Question 1

A hospital records the blood types of 100 patients. If the sample space consists of the four blood types (A, B, AB, O), what is the probability that a randomly selected patient has blood type AB, given that 20 patients have AB blood type?

Accepted Answer

$$P(AB) = \frac{20}{100} = 0.2$$

Question 2

A student rolls two six-sided dice. What is the probability that the sum of the dice is 7?

Accepted Answer

$$P(	ext{sum}=7) = \frac{6}{36} = \frac{1}{6}$$

Question 3

A deck of 52 cards is shuffled. What is the probability of drawing either a heart or a king?

Accepted Answer

$$P(	ext{heart or king}) = \frac{13 + 4 - 1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 4

In a factory, 5 machines are available and 3 must be selected to operate. How many different ways can the machines be chosen?

Accepted Answer

$$\binom{5}{3} = \frac{5!}{3!2!} = 10$$

Question 5

A bag contains 3 red, 2 blue, and 5 green marbles. If one marble is drawn at random, what is the probability it is not green?

Accepted Answer

$$P(	ext{not green}) = \frac{3+2}{10} = \frac{5}{10} = 0.5$$

Question 6

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

Accepted Answer

$$P(	ext{exactly 2 heads}) = \binom{3}{2} \left(\frac{1}{2}\$$right)^2$$ \left(\frac{1}{2}\$$right)^1$$ = 3 	imes \frac{1}{4} 	imes \frac{1}{2} = \frac{3}{8}$$

Question 7

A discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 0.2 + 1.0 + 0.9 = 2.1$$

Question 8

A binomial experiment consists of 8 trials with probability of success $$p=0.3. $$What is the probability of exactly 2 successes?

Accepted Answer

$$P(X=2) = \binom{8}{2} (0.3)^2$ (0.7)^6$ = 28 	imes 0.09 	imes 0.117649 = 0.296$$

Question 9

A normal distribution has mean $$\mu=50$$ and standard deviation $$\sigma=10. $$What is the $$z-$$score for $$x=65$$?

Accepted Answer

$$z = \frac{65-50}{10} = 1.5$$

Question 10

If the area to the left of a $$z-$$score is $$0.975$$, what is the corresponding $$z-$$score?

Accepted Answer

$$z \approx 1.96$$

Question 11

A normal distribution has $$\mu=100$$ and $$\sigma=15. $$What is the probability that a randomly selected value is less than $$115$$?

Accepted Answer

$$z = \frac{115-100}{15} = 1$$; 	ext{Area to the left of } z=1 \approx 0.8413$

Question 12

A student scores 80 on a test where the mean is 70 and the standard deviation is 5. What percentile is the student in?

Accepted Answer

$$z = \frac{80-70}{5} = 2$$; 	ext{Percentile} \approx 97.72\%$

Question 13

A probability distribution for $$X is$$: $$P(X=0)=0.1$$, $$P(X=1)=0.4$$, $$P(X=2)=0.5. $$What is the variance $$\$$sigma^2$$?

Accepted Answer

$$E(X) = 0 	imes 0.1 + 1 	imes 0.4 + 2 	imes 0.5 = 1.4$$; $$\$$sigma^2$$ = (0-1.4)^2$ 	imes 0.1 + (1-1.4)^2$ 	imes 0.4 + (2-1.4)^2$ 	imes 0.5 = 0.196$$

Question 14

A company finds that 60% of its employees commute by car. If 5 employees are randomly selected, what is the probability that exactly 3 commute by car?

Accepted Answer

$$P(X=3) = \binom{5}{3} (0.6)^3$ (0.4)^2$ = 10 	imes 0.216 	imes 0.16 = 0.3456$$

Question 15

If $$P(A)=0.3$$, $$P(B)=0.5$$, and $$P(A \cap B)=0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.5 - 0.2 = 0.6$$

Question 16

A normal distribution has $$\mu=60$$ and $$\sigma=8. $$What value $$x$$ corresponds to $$z=2$$?

Accepted Answer

$$x = \mu + z \sigma = 60 + 2 	imes 8 = 76$$

Question 17

A student randomly selects a letter from the word 'STATISTICS'. What is the probability that the letter is 'S'?

Accepted Answer

$$P(	ext{S}) = \frac{3}{10}$$

Question 18

A medical test correctly identifies a disease 95% of the time. If two patients are tested, what is the probability both tests are correct?

Accepted Answer

$$P(	ext{both correct}) = 0.95 	imes 0.95 = 0.9025$$

Question 19

A probability distribution has $$P(X=1)=0.3$$, $$P(X=2)=0.4$$, $$P(X=3)=0.3. $$What is the standard deviation?

Accepted Answer

$$E(X) = 1 	imes 0.3 + 2 	imes 0.4 + 3 	imes 0.3 = 2.0$$; $$\$$sigma^2$$ = (1-2)^2$ 	imes 0.3 + (2-2)^2$ 	imes 0.4 + (3-2)^2$ 	imes 0.3 = 0.6$$; $$\sigma = \sqrt{0.6} \approx 0.775$$

Exam 2 Study Guide: Probability and Probability Distributions

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