Often it is required to compute the probability of an event given that another event has occurred. Example 1What is the probability that two cards drawn at random from a deck of playing cards will both be aces? It might seem that you could use the formula for the probability of two independent events and simply multiply Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the P(ace on second draw | an ace on the first draw). The vertical bar “|” is read as “given,” so the above expression is short for “The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw.” What is this probability? After an ace is drawn on the first draw, there are 3 aces out of 51 total cards left. This means that the conditional probability of drawing an ace after one ace has already been drawn is Thus, the probability of both cards being aces is Conditional ProbabilityThe probability the event P(B | A) This is read as “the probability of Example 2Find the probability that a die rolled shows a 6, given that a flipped coin shows a head. These are two independent events, so the probability of the die rolling a 6 is Example 3The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:
Notice from the last example that P(B | A) is These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood. Conditional Probability FormulaIf Events P(A and B) = P(A) P(B | A) Example 4If you pull 2 cards out of a deck, what is the probability that both are spades? The probability that the first card is a spade is The probability that the second card is a spade, given the first was a spade, is The probability that both cards are spades is Example 5If you draw two cards from a deck, what is the probability that you will get the Ace of Diamonds and a black card? You can satisfy this condition by having Case A or Case B, as follows:
Let’s calculate the probability of Case A. The probability that the first card is the Ace of Diamonds is Now for Case B: the probability that the first card is black is Recall that the probability of A or B is Try it Now 1In your drawer you have 10 pairs of socks, 6 of which are white. If you reach in and randomly grab two pairs of socks, what is the probability that both are white? Example 6A home pregnancy test was given to women, then pregnancy was verified through blood tests. The following table shows the home pregnancy test results. Find
The second result is what is usually called a false positive: A positive result when the woman is not actually pregnant.
David Lippman, Math in Society, “Probability,” licensed under a CC BY-SA 3.0 license. Step-by-step explanation : Probability in a pack of 52 cards || All basic concepts of cards in probability || class 10 maths
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