Computing Probabilities

STAT 20: Introduction to Probability and Statistics

Adapted by Gaston Sanchez

Review

Conditional Probability

For two events \(A\) and \(B\) we denote the:

• conditional probability of \(A\) given \(B\) as: \(\quad P(A \vert B)\)

• conditional probability of \(B\) given \(A\) as: \(\quad P(B \vert A)\)

If \(A\) and \(B\) are independent, then:

• \(\quad P(A \vert B) = P(A)\)

• \(\quad P(B \vert A) = P(B)\)

Multiplication rule

For two events \(A\) and \(B\) \(\quad \Rightarrow \quad\) \(P(A \text{ and } B) = P(A \vert B) P(B)\)

Complement Rule

\(P(A^C) = 1 - P(A)\)

Concept Question 1

Concept Question 1a

Flip 3 coins, one at a time. Define the following events:

\(A\) is the event that the first coin flipped shows a head

\(B\) is the event that the first two coins flipped both show heads

\(C\) is the event that the last two coins flipped both show tails

What is the outcome space \(\Omega\)?

What is the outcome space \(\Omega\)?

\(A\) is the event that the first coin flipped shows a head

Find \(P(A)\)

\(\Omega\)

  H  H  H
  H  H  T
  H  T  H
  H  T  T
  T  H  H
  T  H  T
  T  T  H
  T  T  T

\(B\) is the event that the first two coins flipped both show heads

Find \(P(B)\)

\(\Omega\)

  H  H  H
  H  H  T
  H  T  H
  H  T  T
  T  H  H
  T  H  T
  T  T  H
  T  T  T

\(C\) is the event that the last two coins flipped both show tails

Find \(P(C)\)

\(\Omega\)

  H  H  H
  H  H  T
  H  T  H
  H  T  T
  T  H  H
  T  H  T
  T  T  H
  T  T  T

Concept Question 1b

Flip 3 coins, one at a time. Define the following events:

\(A\) is the event that the first coin flipped shows a head

\(B\) is the event that the first two coins flipped both show heads

\(C\) is the event that the last two coins flipped both show tails

Events \(A\) and \(B\) are:

1. Independent
2. Mutually exclusive
3. None of the above

After they do this and next, write out the outcome space for the results of flipping 3 coins. And go through what are A, B, and C. A and B are neither independent nor mutually exclusive.

Concept Question 1c

Flip 3 coins, one at a time. Define the following events:

\(A\) is the event that the first coin flipped shows a head

\(B\) is the event that the first two coins flipped both show heads

\(C\) is the event that the last two coins flipped both show tails

Events \(A\) and \(C\) are:

1. Independent
2. Mutually exclusive
3. None of the above

After they do this and next, write out the outcome space for the results of flipping 3 coins. And go through what are A, B, and C. A and C are independent. Show that P(A & C ) = 1/8 = P(A)P(C) = 1/4 * 1/2

Concept Question 2

Concept Question 2

Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}.

• Let \(A\) be the event that at least one of the tickets drawn is labeled with an even number.

• Let \(B\) be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number).

Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}.

What is the outcome space \(\Omega\)?

Let \(A_1\) be the event that the 1st ticket is an even #

Find \(P(A_1)\)

Let \(A_2\) be the event that the 2nd ticket is an even #

Find \(P(A_2)\)

Let \(A\) be the event that at least one of the tickets is an even #

Find \(P(A)\)

Let \(A\) be the event that at least one of the tickets is an even #

Find \(P(A) = 1 - P(A^c)\)

Let \(B\) be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number).

Find \(P(B)\)

  Prime numbers:  2, 3, 5, 7
  Non-prime #'s:  1, 4, 6, 8, 9

Let \(B\) be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number).

Find \(P(B) = 1 - P(B^c)\)

  Prime numbers:  2, 3, 5, 7
  Non-prime #'s:  1, 4, 6, 8, 9

Concept Question 2b

Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}.

Let A be the event that at least one of the tickets drawn is labeled with an even number.

Let B be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number).

Suppose the numbers on the tickets drawn are 3 and 9.

Which of the following events occur?

1. \(A\)

2. \(B\)

3. \(A\) and \(B\) (\(A \cap B\))

4. \(A\) and \(B^C\)

5. \(A^C\) and \(B\)

ii and v, I told them they can immediately see that A^C is true since no even number, so definitely (v) and then we do have a prime number and both are odd so (ii) is true.

03:00

Concept Question 3

Concept Question 3a

02:00

The 2024 World Series was the championship series was a best-of-seven playoff between the Los Angeles Dodgers and the New York Yankees. The winners in the World Series have to win a majority of 7 games, so the first team to win 4 games wins the series.

Suppose we assumed that the probability that the Dodgers would have beaten the Yankees in any single game was estimated at 56%, independently of all the other games.

What was the probability that the Dodgers would have won in a clean sweep?

Straightforward application of multiplication rule for independent events. probability is 0.6^4 = 0.1296. Follow up comment/question: if the teams were more evenly matched, say probability of either team winning is 50%, the probability of going to 7 games and probability of winning in 6 games is equal given 5 games are played.

Concept Question 3b

02:00

Suppose we assume, instead, that the probability that the Dodgers would have beaten the Yankees in any single game was 50%, independently of all the other games.

In this case, was the probability that the series would have gone to 6 games higher than the probability that the series would have gone to 7 games, given that 5 games were played?

if 5 games are played it means one of the teams is leading 3-2. Therefore there are two scenarios: the team that is ahead wins game 6 and the series, or the team that is behind wins game 6 and they go to 7 games. Both are equally likely.

tinyurl.com/yujrzpm9

Practice Problems

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Worksheet

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