Problem 25 - Calculus for Biology and Medicine

Question

A bag contains two coins, one fair and the other with two heads. You pick one coin at random and flip it. Find the probability that the outcome is heads.

Step-by-Step Solution

Answer

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

Step 1: Identify Outcomes

In this problem, we have two coins: one fair and the other with two heads. Therefore, when we pick a coin at random, there are two possible outcomes. The first coin (fair coin) will have a head on one side and a tail on the other, while the second coin (two-headed coin) will have heads on both sides.

Step 2: Calculate Basic Probabilities

The probability of picking the fair coin is 1/2, and the probability of picking the two-headed coin is also 1/2 because there are only two coins and no preference for either.

Step 3: Calculate Conditional Probabilities

If the fair coin is picked, the probability of getting heads is 1/2, since it's equally likely to land on heads or tails. If the two-headed coin is picked, the probability of getting heads is 1 because both sides are heads.

Step 4: Use Law of Total Probability

To find the probability of flipping heads, sum the probabilities of flipping heads when each coin is selected, weighted by the probability of picking that coin. Using the law of total probability, the calculation is as follows:\[P(H) = P(H | C_1) \cdot P(C_1) + P(H | C_2) \cdot P(C_2) = \left( \frac{1}{2} \cdot \frac{1}{2} \right) + (1 \cdot \frac{1}{2})\]Where:- \(C_1\) is the event of picking a fair coin- \(C_2\) is the event of picking a two-headed coin- \(P(H | C_1)\) is the probability of heads if the fair coin is chosen- \(P(H | C_2)\) is the probability of heads if the two-headed coin is chosen.

Step 5: Calculate and Simplify

Calculate the total probability:\[P(H) = \left( \frac{1}{2} \cdot \frac{1}{2} \right) + (1 \cdot \frac{1}{2}) = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\]

Step 6: Conclusion

The probability that the outcome is heads when a coin is picked at random and flipped is \( \frac{3}{4} \).

Key Concepts

Understanding Conditional Probability

Conditional probability is a fundamental concept in probability theory. It helps us understand the likelihood of an event happening based on the occurrence of a previous event.
In simple terms, it's about revising probabilities in light of additional evidence.
For instance, in our exercise, once we know which type of coin we've picked, we change our probability estimates about flipping heads.
- If we pick the fair coin, there's a 50% chance each for heads or tails.- Pick the two-headed coin, and heads become certain, a 100% event.

This adjustment based on the coin choice is the essence of conditional probability. It's symbolically denoted by \( P(A|B) \), meaning the probability of event \( A \) occurring given that \( B \) is true. Here, \( A \) is flipping heads, and \( B \) is picking a specific coin.

Employing the Law of Total Probability

The law of total probability is pivotal in scenarios with multiple pathways leading to the same outcome.
This law allows us to calculate the overall probability of an event by considering all possible ways it can happen and weighting each by the probability of the condition leading to it.
- In the exercise, the law helps us compute the probability of getting heads by considering both possibilities: choosing the fair coin and choosing the two-headed coin.We calculate this by:\[P(H) = P(H \mid C_1) \cdot P(C_1) + P(H \mid C_2) \cdot P(C_2)\]Where:

• \( C_1 \) is selecting the fair coin.
• \( C_2 \) is selecting the two-headed coin.

Both contribute to getting heads, with their overall impact depending on how likely each coin is picked. Hence, the law of total probability integrates these separate paths into a unified answer.

Exploring Random Variables

Random variables transform potential outcomes of experiments into numerical values. They're used to describe outcomes of random processes, like flipping a coin.
In the given exercise, each coin flip is an example of a random variable because it can yield different results depending on the coin selected. For the fair coin:

• The random variable could be 0 for tails and 1 for heads.

In the case of the two-headed coin:

• The random variable always returns 1, because every flip results in heads.

Understanding random variables helps us link abstract probability problems with concrete numerical outcomes.
This way, we can systematically analyze the results of our random decisions, like choosing which coin to flip.