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

When a coin is tossed three times, the total number of possibilities can be represented as combinations of H (head) and T (tail). Thus the sample space consists of the following eight outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 2: Identify the Condition

We are given the condition that the second toss results in a head (H). This condition reduces our sample space. Only the outcomes where the second toss is a head are: HHH, HHT, THH, THT.

Step 3: Define the Event of Interest

The event we are interested in is getting at least two heads. From the reduced sample space (HHH, HHT, THH, THT), the outcomes that satisfy the condition of at least two heads are: HHH, HHT, THH.

Step 4: Calculate the Conditional Probability

The probability we need is the ratio of the number of favorable outcomes (with at least two heads) to the number of possible outcomes (given that the second toss is a head). There are 3 favorable outcomes (HHH, HHT, THH) and 4 possible outcomes (HHH, HHT, THH, THT). Thus, the probability is \( \frac{3}{4} \).

Key Concepts

Sample Space

In probability, the term "sample space" refers to the set of all possible outcomes that can result from a particular experiment or event. When tossing a coin three times, each toss can result in either a head (H) or a tail (T). This gives us a sequence of combinations for multiple tosses.

For example, if a coin is tossed three times, each outcome represents a sequence of heads and tails. Thus, the complete sample space for tossing a coin three times would include eight outcomes:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

This complete set of eight outcomes ensures that every possible scenario is considered, forming the full sample space for this experiment.

Coin Toss Outcomes

When analyzing coin toss outcomes, especially under specific conditions, certain sequences become more relevant. In this exercise, we are interested in outcomes when tossing a fair coin three times, but with an added condition.

The condition stated here is that the second toss must be a head (H). This condition effectively reduces the sample space by eliminating sequences that have a tail on the second toss. The outcomes that meet this condition are:

• HHH
• HHT
• THH
• THT

These refined results focus only on the possibilities where the second toss results in heads, thereby narrowing down the potential outcomes we need to consider for subsequent calculations.

Probability Calculation

Calculating conditional probability involves comparing the number of favorable outcomes to the total number of plausible outcomes meeting a specific criterion. In our scenario, we are interested in the probability of getting at least two heads, given that the second toss is a head.

From the reduced sample space (HHH, HHT, THH, THT), we first identify which outcomes fulfill the requirement of having at least two heads:

• HHH
• HHT
• THH

Here, there are 3 favorable outcomes. Since there were originally 4 possible outcomes considering the condition (the complete reduced sample space), the conditional probability can be calculated as follows: The probability is \[ \frac{3}{4} \]This fraction indicates that there is a 75% chance of having at least two heads when it's known that the second toss results in a head.