Question

If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.

Short answer

The conditional probability of getting five heads given that there are at least four heads is \( \frac{(0.5)^5} {(0.5)^4 + (0.5)^5} \)

Step-by-step solution

Understand and Define the Events

Let's define the events A and B. Event A is the chance of getting five heads (HHHHH) when five independent coin tosses are made. Event B is getting at least four heads out of five coin tosses, that is getting either four heads out of five tosses or five heads.

Find the Probability of Event A

The probability of getting 5 heads (Event A) is calculated as \(P(A) = (0.5)^5\), since the chance of getting head in each independent toss is 0.5.

Find the Probability of Event B

The probability of getting 'at least four heads' (Event B) is calculated as \(P(B) = (0.5)^4 + (0.5)^5\), since getting at least four heads (HHHHT,HHHTH, etc.) implies getting either exactly four heads or five heads. Each outcome is mutually exclusive, which allows us to calculate each case separately and then add the results.

Find the Conditional Probability of A given B

To find the conditional probability of A given B, apply the formula P(A|B) = P(A ∩ B) / P(B). But here A is a subset of B, so P(A ∩ B) is simply P(A). So, P(A|B) = P(A) / P(B) = \( \frac{(0.5)^5} {(0.5)^4 + (0.5)^5} \)

Key concepts

Independent Events Probability

When we talk about independent events in probability, we're referring to a scenario where the outcome of one event has no effect on the outcome of another. In other terms, the occurrence of one event does not change the probability of the other event occurring. A classic example of independent events is flipping a coin. The result of a coin toss (heads or tails) does not influence the result of the subsequent tosses.

Each coin toss is an independent event with its own 50-50 chance of landing on heads or tails, often represented as a probability of 0.5 for each outcome. When multiple independent events happen one after another, like flipping a coin five times, their probabilities are multiplied to calculate the combined outcome's probability. For example, the probability of getting heads on five consecutive coin tosses would be calculated as the product of each individual event's probability, or \( (0.5)^5 \).

Probability of Coin Toss Outcomes

The probability of specific outcomes from flipping a coin multiple times can be quite intriguing to calculate. Since each toss of a fair coin has only two possible outcomes – heads (H) or tails (T) – the probability of each outcome is 0.5. If we were to toss a coin twice, we'd have four possible outcomes: HH, HT, TH, TT.

Now, considering independent coin tosses multiple times, like five tosses, we can find probabilities of particular events, such as 'getting at least four heads' by considering all the relevant combinations that satisfy the condition (four heads plus one tail, or five heads). For instance, the probability of getting exactly four heads in five tosses involves finding the probability for each unique arrangement of four heads and one tail (e.g., HHHHT, HHHTH, etc.) and summing those probabilities together.

Conditional Probability Formula

Conditional probability is the probability of an event occurring given that another event has already occurred. The formula used to calculate this is \( P(A|B) = P(A \cap B) / P(B) \), where \( P(A|B) \) is the probability of event A occurring given that event B has occurred, and \( P(A \cap B) \) is the probability that both events A and B occur.

In cases where event A is completely contained within event B (like getting five heads in five coin tosses is contained within getting at least four heads), \( P(A \cap B) \) is simply \( P(A) \). Thus, the conditional probability simplifies to \( P(A)/P(B) \). This formula helps us understand the likelihood of a particular outcome within a subset of conditions and is a fundamental concept in probability theory.