Combined Events and Conditional ProbabilityAQA GCSE Maths: Flashcards

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What is the formula for calculating the probability of two independent events A and B both occurring?

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What is the formula for calculating the probability of two independent events A and B both occurring?: P(A and B) = P(A) × P(B) This formula only applies when events are independent (the outcome of one event does not affect the outcome of the other).
Define independent events in probability.: Independent events are two or more events where the outcome of one event does not affect the probability of the other event occurring. The events are unrelated in terms of their outcomes.
Define dependent events in probability.: Dependent events are two or more events where the outcome of one event affects the probability of the other event occurring. They are related, and the probability of the second event changes based on the first event's outcome.
What is a tree diagram used for in probability?: A tree diagram is used to show all possible outcomes of two or more combined events. It displays each event as branches with probabilities labelled on each branch, allowing you to calculate probabilities of combined events by multiplying along branches.
Explain how to calculate the probability of combined events using a tree diagram.: Multiply the probabilities along each branch to find the probability of that particular outcome. For example, if Event A has probability 0.6 and Event B has probability 0.5, the probability of both occurring is 0.6 × 0.5 = 0.3. Add the probabilities of branches if you want the probability of different outcomes.
What is conditional probability?: Conditional probability is the probability of an event occurring given that another event has already occurred. It shows how the probability of one event changes when we know that another event has happened.
How can tree diagrams be used to find conditional probabilities?: On a tree diagram, the second set of branches shows conditional probabilities that depend on what happened in the first event. The probabilities on these branches may be different from the first event's branches because they are conditional on the first outcome.
How can Venn diagrams be used to show conditional probability?: Venn diagrams display overlapping sets for events A and B. Conditional probability P(A|B) is calculated as the number of outcomes in both A and B divided by the total number of outcomes in B. This shows the probability of A given that B has occurred.
State the addition law for probability.: P(A or B) = P(A) + P(B) – P(A and B) You subtract P(A and B) because this outcome is counted in both P(A) and P(B), so you must remove the overlap to avoid double-counting.
When calculating P(A or B), why do you subtract P(A and B)?: Because the outcomes that are in both A and B are included in both P(A) and P(B). If you only added them, you would count these overlapping outcomes twice, so you must subtract them once to get the correct probability.
What is a two-way table used for in probability?: A two-way table (contingency table) displays the frequencies or outcomes for two events in a grid format. It allows you to easily calculate probabilities by reading off the relevant cells and dividing by the total number of outcomes.
Explain how to calculate probabilities using a two-way table.: Find the relevant cell(s) containing the outcome(s) you want. Divide the frequency in that cell by the total number of all outcomes in the table. For combined events, add relevant cells before dividing by the total.
How do you determine if events are independent using their probabilities?: Events A and B are independent if P(A and B) = P(A) × P(B). If this equation is true, the events are independent. If the equation is false, the events are dependent.
When working with dependent events, how do the probabilities on a tree diagram change?: The probabilities on the second set of branches differ from the first set because they depend on the outcome of the first event. For example, when selecting items without replacement, the total number of items decreases, so the probabilities change for the second selection.
What is the difference between P(A and B) for independent versus dependent events?: For independent events: P(A and B) = P(A) × P(B) For dependent events: P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A has already occurred. The second probability is conditional on the first event.