Counting and Probability

1.

A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, blueberry, and pistachio ice cream, served on either a waffle, sugar, or plain cone. How many different single-scoop ice-cream cones can you buy from this vendor?

🔗

Your answer is :

🔗

Answer.

\(15\)

🔗

2.

A girl has 8 skirts, 9 blouses, and 9 pairs of shoes. How many different skirt-blouse-shoe outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.)

🔗

Your answer is :

🔗

Answer.

\(648\)

🔗

3.

Standard automobile license plates in a country display 2 numbers, followed by 2 letters, followed by 3 numbers. How many different standard plates are possible in this system? (Assume lead 0 is not allowed, but repetitions of letters and numbers are allowed.)

🔗

Your answer is :

🔗

Answer.

\(60840000\)

🔗

4.

A true-false test contains 23 questions. In how many different ways can this test be completed. (Assume we don’t care about our scores.)

🔗

Your answer is :

🔗

Answer.

\(8388608\)

🔗

5.

In how many ways can first, second, and third prizes be awarded in a contest with 970 contestants?

🔗

Your answer is :

🔗

Answer.

\(909852240\)

🔗

6.

How many ways are there to seat 3 people in a row of 5 chairs?

🔗

Answer.

\(60\)

🔗

Solution.

\({5 \choose 3} = \frac{5 !}{3 ! 2!}\)

\begin{equation*} {5 \choose 3} \cdot 3 ! =\frac{5 !}{3 ! 2!} \cdot 3! = \frac{5 !}{2!} = 60 \end{equation*}

\(\frac{5 !}{2!} = 60\)

🔗

7.

An American roulette wheel has 38 slots: two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot.

🔗

Your answer is :

🔗

Answer.

\(0.473684210526316\)

🔗

8.

In the 4/41 lottery game, a player selects 4 numbers from 1 to 41. What is the probability of picking the 4 winning numbers?

🔗

Your answer is :

🔗

Answer.

\(9.87459267305224\times 10^{-6}\)

🔗

9.

Two marbles are drawn randomly one after the other without replacement from a jar that contains 5 red marbles, 4 white marbles, and 7 yellow marbles. Find the probability of the following events.

🔗

(a) A red marble is drawn first followed by a white marble.

🔗

The probability is :

🔗

(b) A white marble is drawn first followed by a white marble.

🔗

The probability is :

🔗

(c) A yellow marble is not drawn at all.

🔗

The probability is :