Conditional Probability

What is conditional probability?

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is denoted by P(A|B), where A is the event that has already occurred and B is the event that we are interested in the probability of.

For example, if you flip a coin and then roll a die, the probability of getting heads on the coin, given that you rolled a 6 on the die, is 1/2. This is because there is still a 1/2 chance of getting heads on the coin, regardless of what happened on the die.

Mathematical definition of conditional probability

The mathematical definition of conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

where:

  • P(A|B) is the probability of event A occurring, given that event B has already occurred
  • P(A ∩ B) is the probability of events A and B both occurring
  • P(B) is the probability of event B occurring

For example, if you flip a coin and then roll a die, the probability of getting heads on the coin, given that you rolled a 6 on the die, is:

P(A|B) = P(A ∩ B) / P(B) = (1/2) / (1/6) = 1/2

Multiple choice questions on conditional probability

Here are some multiple choice questions on conditional probability with answers:

  1. What is the probability of getting heads on a coin, given that you rolled a 6 on a die?
    • 1/2
    • 1/3
    • 2/3
    • 3/4
    • The answer is 1/2. There are two possible outcomes when you flip a coin, heads or tails. The probability of getting heads is still 1/2, regardless of what happened on the die.
  2. What is the probability of rolling a 6 on a die, given that you got heads on a coin?
    • 1/2
    • 1/3
    • 2/3
    • 3/4
    • The answer is 1/3. There are 6 possible outcomes when you roll a die, but only 1 outcome in which you get a 6 and heads (6H). The probability of rolling a 6 and getting heads is 1/6 / 1/2 = 1/3.
  3. What is the probability of getting heads on a coin, given that you rolled an even number on a die?
    • 1/2
    • 1/3
    • 2/3
    • 3/4
    • The answer is 2/3. There are 3 possible outcomes when you roll a die that are even: 2, 4, and 6. There are also 2 possible outcomes when you flip a coin: heads and tails. The probability of getting heads and an even number on the die is 2/3 / 2 = 2/3.