Probability of combined events Revision Notes

Combined events involve two or more events happening in sequence or together. When events are considered with replacement, it means after the first event occurs, the situation resets to the original state before the next event happens. For example, if you draw a card from a deck and then replace it before drawing again, the probabilities for the second draw remain the same as the first.

This assumption of replacement ensures that the events are independent; the outcome of one event does not affect the probabilities of the next. This is important when calculating combined probabilities because it allows multiplication of individual event probabilities directly.

• Replacement means the sample space remains unchanged between events.
• Events with replacement are independent events.
• Probability of combined events is found by considering all possible outcomes.

Sample space diagrams list all possible outcomes of combined events in a grid or table format. Each cell represents one possible combined outcome. When events are with replacement, the sample space for each event remains the same.

For example, consider tossing a fair coin twice with replacement (the coin is tossed twice independently). The sample space diagram is:

First toss outcomes: H or T; Second toss outcomes: H or T; Combined outcomes: HH, HT, TH, TT. Each outcome is equally likely with probability 1/4 (since 1/2 × 1/2 = 1/4).

• List all possible combined outcomes in a grid.
• Each outcome's probability is the product of individual event probabilities.
• Sum probabilities of favorable outcomes to find combined event probability.

Venn diagrams visually represent two events and their relationships, including intersections and unions. They are useful for showing probabilities of combined events involving two sets.

For example, if event A and event B are two events with probabilities P(A) and P(B), and P(A ∩ B) is the probability both occur, the Venn diagram shows overlapping circles representing these events.

When events are with replacement and independent, the intersection probability is P(A) × P(B). The Venn diagram helps identify probabilities of A only, B only, both, or neither by shading areas accordingly.

• Use two overlapping circles to represent two events.
• Label probabilities inside and outside the intersection.
• Calculate combined probabilities using areas of overlap and union.
• Limited to two events for clarity and accuracy.

Tree diagrams are ideal for sequential combined events, especially with replacement. Each branch represents a possible outcome of an event, with probabilities written alongside.

For example, tossing a fair coin twice: the first branches are H (probability 1/2) and T (1/2). From each of these, two branches again for the second toss: H (1/2) and T (1/2). Outcomes at the ends are HH, HT, TH, TT.

To find the probability of a combined outcome, multiply the probabilities along the branches. For example, P(H then T) = 1/2 × 1/2 = 1/4. Summing probabilities of favorable outcomes gives the total probability for the combined event.

• Draw branches for each event outcome with probabilities.
• Multiply probabilities along branches to find combined outcome probability.
• Sum relevant outcomes for total combined event probability.
• Clearly label outcomes at branch ends.

Once all possible combined outcomes are identified using diagrams, calculate the probability of each outcome. For independent events with replacement, multiply the probabilities of individual events.

To find the probability of a specific combined event (e.g., getting at least one head in two coin tosses), sum the probabilities of all favorable outcomes.

Example: Probability of at least one head in two coin tosses = P(HH) + P(HT) + P(TH) = 1/4 + 1/4 + 1/4 = 3/4.

• Multiply individual event probabilities for combined outcomes.
• Sum probabilities of all favorable outcomes.
• Check that total probabilities sum to 1 across the entire sample space.