Q: A sandwich shop offers 3 types of bread, 2 types of cheese, 4 types of meat, and 3 types of sauce. If a customer selects one of each, how many unique sandwiches can be made?

$$3 \times 2 \times 4 \times 3 = 72$$

Question 1

A sandwich shop offers 3 types of bread, 2 types of cheese, 4 types of meat, and 3 types of sauce. If a customer selects one of each, how many unique sandwiches can be made?

Accepted Answer

$$3 	imes 2 	imes 4 	imes 3 = 72$$

Question 2

How many ways are there to select 6 students from a group of 16 to form a committee?

Accepted Answer

$$16C6 = 8,008$$

Question 3

A delivery driver must visit 3 locations out of 9 remaining. In how many different orders can the driver make these deliveries?

Accepted Answer

$$9P3 = 504$$

Question 4

A thief tries to guess a 3-digit PIN code using a keypad with 7 keys (digits 1-7). Digits can repeat. What is the probability of guessing the correct code on the first try?

Accepted Answer

$$\frac{1}{343}$$

Question 5

You visit 12 colleges and want to apply to 4 of them. In how many ways can you choose the 4 colleges to apply to?

Accepted Answer

$$12C4 = 495$$

Question 6

You want to rank your top 5 colleges out of 12 you visited. How many different rankings are possible?

Accepted Answer

$$12P5 = 665,280$$

Question 7

Given the following probability distribution for a random variable $$X$$:

$$
\begin{align*}
x & : 0 & 1 & 2 & 3 & 4 & 5 \
P(x) & : 0.025 & 0.164 & 0.311 & 0.311 & 0.164 & 0.025 \
\end{align*}

Is $$this a valid probability distribution, and if so, what is the mean $$\mu_X$$?

Accepted Answer

Yes, it is a probability distribution; $$\mu_X = 2.5$$

Question 8

Which of the following is NOT a requirement for a probability distribution?

Accepted Answer

The mean of the distribution must be zero

Question 9

A Gallup poll found that 12% of adults believe it is bad luck to walk under a ladder. If 30 adults are randomly selected, what is the probability that at most 5 of them hold this belief? (Let $$X \sim 	ext{Bin}(30, 0.12))$$

Accepted Answer

$$P(X \leq 5) = 0.8559$$

Question 10

Which of the following scenarios does NOT result in a binomial distribution?

Accepted Answer

Surveying 100 students and recording only whether each likes pirate or ninja movies (Yes/No), but not counting the number of Yes responses

Question 11

Suppose 44% of adults with smartphones use them in meetings or classes. If 20 adults are randomly selected, what is the probability that exactly 15 use their smartphones in meetings or classes? (Let $$X \sim 	ext{Bin}(20, 0.44))$$

Accepted Answer

$$P(X = 15) = 0.0038$$

Question 12

Suppose $$X \sim 	ext{Bin}(20, 0.44). $$What is the probability that at most 10 adults use their smartphones in meetings or classes?

Accepted Answer

$$P(X \leq 10) = 0.7788$$

Question 13

A random variable $$X$$ has the following probabilities: $$P(0) = 0.024$$, $$P(1) = 0.164$$, $$P(2) = 0.309$$, $$P(3) = 0.308$$, $$P(4) = 0.166$$, $$P(5) = 0.026. Is $$this a valid probability distribution?

Accepted Answer

No, because the probabilities do not sum to 1

Question 14

Which formula would you use to calculate the number of ways to arrange 5 out of 12 colleges in order of preference?

Accepted Answer

$$12P5$$

Question 15

A geneticist is studying an X-linked disorder. If 5 males with the disorder each have one child, and the probability distribution for the number of children inheriting the disorder sums to 1, what does this indicate?

Accepted Answer

It is a valid probability distribution

Question 16

Which of the following best describes the difference between a combination and a permutation?

Accepted Answer

A combination does not consider order, while a permutation does

Question 17

If you are choosing 4 colleges to apply to from 12, and the order does not matter, which formula should you use?

Accepted Answer

$$12C4$$

Question 18

A binomial experiment consists of 30 trials with probability of success $$p = 0.12. $$Which formula gives the probability of exactly $$k$$ successes?

Accepted Answer

$$P(X = k) = \binom{30}{k} (0.12)^k (0.88$$)^{30-k}$$

Step-by-Step Guidance for Statistics Test 3 (Combinatorics, Probability, and Binomial Distributions)

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