Problem 12 - Calculus for Biology and Medicine

Question

You toss a fair coin three times. Find the probability that at least two heads occurred given that the second toss resulted in heads.

Step-by-Step Solution

Answer

The probability is \( \frac{3}{4} \).

Step 1: Define the Sample Space

First, identify all possible outcomes when a coin is tossed three times. Each toss has two possible results: heads (H) or tails (T). Thus, for three tosses, the sample space consists of 2^3 = 8 outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 2: Apply the Given Condition

Since the second toss is given to be heads, we only consider outcomes where the second coin is H. These are: HHH, HHT, THH, and THT. Hence, the reduced sample space, considering the condition, has 4 outcomes.

Step 3: Identify Favorable Outcomes

We need at least two heads. Looking at the reduced sample space (HHH, HHT, THH, THT), the outcomes containing at least two heads are HHH, HHT, and THH. Thus, there are 3 favorable outcomes.

Step 4: Calculate the Probability

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in the conditional sample space. \[P(\text{at least two heads} \mid \text{second toss is heads}) = \frac{3}{4}\]

Key Concepts

Conditional Probability

Conditional probability can be thought of as the probability of one event occurring given that another event has already occurred. It's like having a clue or piece of information that allows you to narrow down the possibilities. In the coin toss scenario, we know the second toss is heads. This information shapes the probability of getting at least two heads. We denote conditional probability using the notation: - \( P(A \mid B) \), meaning "probability of event A given event B." - To find the conditional probability, you focus only on the outcomes that satisfy the given condition (in this case, the second toss being heads), and determine the likelihood of the event of interest (getting at least two heads) within this narrowed down set.

Sample Space

The sample space is essentially the entire set of all possible outcomes of an experiment. For our three coin tosses, the sample space starts by including every combination of heads and tails possible with three coins; that's 2 choices (heads or tails) for each toss, so altogether there are 2 raised to the power of 3, or 8 possible combinations. - These are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. - However, when a condition is imposed (like the second coin being heads), the effective sample space changes. Now it only includes outcomes that meet this condition: HHH, HHT, THH, and THT.

Favorable Outcomes

After determining the valid sample space based on conditions, we identify the favorable outcomes. These are outcomes where our event of interest happens. In this problem, we seek outcomes with at least two heads. Checking the conditional sample space (HHH, HHT, THH, THT), only three outcomes meet the criteria: HHH, HHT, and THH. - Favorable outcomes help in determining probability, as they are used as the numerator in the probability fraction formula. Recognizing these outcomes is crucial for accuracy.

Calculating Probabilities

Once you have the favorable outcomes and the redefined sample space, you can easily calculate probabilities. Probability is the ratio of favorable outcomes to the total number of possible outcomes, in this case, within the sample conditioned on the prior event. - So, the probability that at least two heads occurred given the condition that the second toss was heads is calculated as:\[ P(\text{at least two heads} \mid \text{second toss is heads}) = \frac{3}{4} \]- Here, 3 is the number of favorable outcomes (HHH, HHT, and THH) and 4 is the number of outcomes in the conditional sample space where the second toss is heads. This formula gives you a straightforward path to the solution.