5.E: Basic Concepts of Probability (Exercises)

Exercise \(\PageIndex{1}\): Counting

A typical PIN is a sequence of any seven symbols chosen from the 26 letters in the alphabets and the ten digits \(0, 1,....,9\) with repetition allowed.

1. How many PIN's are available?
2. Supposing that symbols cannot be repeated, then how many PIN's are available?

Exercise \(\PageIndex{2}\): Counting

How many (Canadian) postal codes would there be possible without repetition of letters or numbers?

Exercise \(\PageIndex{3}\): Counting

Seven women and nine men are on the faculty in the Mathematics Department.

1. How many different committees are there which are made up of five members of the department, at least one of which is a woman?
2. How many ways can the five-person committee be arranged around a circular table?

Exercise \(\PageIndex{4}\): Probability

You have 23 people in a room. What is the probability that the two of them have the same birthday?

Exercise \(\PageIndex{5}\):Probability

A coin is tossed \(7\) times, find

1. the probability that we see at least \(2\) heads?
2. the probability that we see exactly \(2\) heads?
3. the probability that we see exactly \(2\) heads or exactly \(4\) heads?

Exercise \(\PageIndex{6}\):Probability

What is the conditional probability that the sum of the dice is 10, 11, or 12 given that the first die rolled comes up a 6?

Exercise \(\PageIndex{7}\):Probability

We have a standard deck of 52 cards and you select a card at random.

1. What is the probability you select a 5?
2. What is the probability you don’t select a 5?
3. What is the probability you select a 5 or 7? Are these mutually exclusive events or not?
4. What is the probability you select a 5 or a diamond? Are these mutually exclusive events or not?
5. What is the probability you select the Jack of Spades given you have selected a face card?
6. What is the probability you select a diamond given you have selected a red card?
7. What is the probability you selected a red card given you’ve selected a diamond?

Now suppose you select two cards from the deck.

8. What is the probability you select two Jacks assuming you replace a card and reshuffle the deck before selecting again? Are these dependent or independent events?
9. What is the probability you select two Jacks assuming you don’t replace a card once you’ve selected it? Are these dependent or independent events?
10. What is the probability you select a Jack and an Ace – again assuming replacement. Careful- you could select the Jack then the Ace or the Ace then the Jack.

Exercise \(\PageIndex{8}\): Probability

In a town of 10,000 people, 400 have beards (all men), 4000 are adult men, and 5 of the townspeople are murderers. All 5 murderers are men and 4 of the murderers have beards.

Suppose you go to this town and select a towns person at random.

• Let A be the event that the person turns out to be one of the five murderers.
• Let B be the event the person is bearded.
• Let C be the event the person is an adult male.

Find P(A), P(A given B), P(B given A), P(A given not B), P(A given C), P(A given not C).

Exercise \(\PageIndex{9}\): False Positive

Disease X is a disease affecting about 1 percent of the population. A test for Disease X will test positive on all afflicted with the disease and will also test positive for 5% of the population who do not have the disease.

1. What is the probability a randomly chosen person has disease X?
2. What is the probability a randomly chosen person will test positive for the disease?
3. Suppose you test positive for the disease. What is the probability you don’t actually have the disease? (This is the conditional probability that you don’t have the disease given you tested positive for it).
4. What would be the probability that you test positive for the disease twice given that you don’t have disease X?
5. Can you identify (at least 2) criticism’s of our theoretical probability calculations here?

Exercise \(\PageIndex{10}\): Probability

1. the probability that we see at least \(2\) heads?
2. the probability that we see exactly \(2\) heads?
3. the probability that we see exactly \( 2 \) heads or exactly \( 4 \) heads?

Exercise \(\PageIndex{11}\): Probability

A multiple-choice test has 10 questions, each with 4 possible answers. A student guesses all ten questions.

1. Find the probability that the student will get all ten questions right.
2. Find the probability that the student will get at least 1 question right.

Exercise \(\PageIndex{12}\): Probability

Suppose two fair dice are thrown.

1. What is the probability the sum of the dice is at most 4?
2. What is the probability the sum of the dice is more than 4?
3. What are the odds the sum of the dice is at most 4?
4. What is the probability the sum of the dice is at most 4 given that the first die shows a 3?

Exercise \(\PageIndex{13}\): Combination

If \(n \geq k+2\) and \(k \geq 2\), show that \( {n \choose k } - {n-2 \choose k} - {n-2 \choose k-2}\) is even.

Exercise \(\PageIndex{14}\): Combination

Find the coefficient of \(x^{5}\) in the binomial expansion of
\[\left( \frac{2}{x}+x^2 \right)^{25}\].

Exercise \(\PageIndex{15}\): Combination

For natural numbers \(n\) and \(r\), \(r<n\), show that

\[\displaystyle {n+1 \choose r } = {n \choose r} + {n \choose r-1}. \]

Answer

under construction.

Exercise \(\PageIndex{16}\):

A fair coin, a double-headed coin and a double-tailed coin are placed in a bag. A coin is randomly selected. The coin selected is then tossed.

1. Find the probability that the coin lands with a “head”.

2. When the coin is tossed, it lands “tail”. Find the probability that it is the double-tailed coin.

Answer

\(\dfrac{1}{2} \dfrac{3}{4}\).