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Mathematics

Probability

Conditional Probability

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What is the probability that a student passes the exam, given that they have studied for over 10 hours?

P(\text{Pass} | \text{Studied} > 10 \text{ hrs}) = \text{Use statistical data from previous exams to calculate this.}

What is the probability that a person is a smoker, given that they have lung cancer?

P(\text{Smoker} | \text{Lung cancer}) = \text{Use medical statistics to calculate this.}

What is the probability of it raining today, given that it rained yesterday?

P(\text{Rain today} | \text{Rain yesterday}) = \text{Use weather statistics and history to calculate this.}

What is the probability of choosing a chocolate donut, given that you are at a bakery that has 3 kinds of donuts (chocolate, vanilla, strawberry) with equal amounts?

P(\text{Chocolate} | \text{at bakery}) = \frac{1}{3}

What is the probability of getting heads on a coin flip, given that the last five flips resulted in heads?

P(\text{Heads} | 5 \times \text{Heads}) = \frac{1}{2}

What is the probability of drawing a red card, given that the card is a 6?

P(\text{Red} | 6) = \frac{P(\text{Red} \cap 6)}{P(6)} = \frac{1/26}{1/13} = \frac{1}{2}

What is the probability of drawing an ace from a deck of cards, given that a heart has been drawn?

P(\text{Ace} | \text{Heart}) = \frac{P(\text{Ace} \cap \text{Heart})}{P(\text{Heart})} = \frac{1/52}{1/4} = \frac{1}{13}

What is the probability of drawing a king, given that a face card has been drawn?

P(\text{King} | \text{Face card}) = \frac{P(\text{King} \cap \text{Face card})}{P(\text{Face card})} = \frac{4/52}{12/52} = \frac{1}{3}

What is the probability of someone being left-handed, given that they are male?

P(\text{Left-handed} | \text{Male}) = \text{Use population statistics to calculate this.}

What is the probability of rolling a 6 on a die, given that an even number has been rolled?

P(6 | \text{Even}) = \frac{P(6 \cap \text{Even})}{P(\text{Even})} = \frac{1/6}{1/2} = \frac{1}{3}

What is the probability that the next traffic light will be green, given that the last two were red?

P(\text{Green} | 2 \times \text{Red}) = \text{Assuming independent events, the probability is simply the regular probability of a green light.}

What is the probability of an individual having a birthday on a weekend, given that they were born in a leap year?

P(\text{Weekend} | \text{Leap Year}) = \frac{P(\text{Weekend})}{\text{Days in Leap Year}} = \frac{104/366}{1} = \frac{104}{366}

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