Problem 49 - A First Course in Probability

Question

When coin 1 is flipped, it lands on heads with probability \(.4 ;\) when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times. (a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips? (b) Given that the first of these ten flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?

Step-by-Step Solution

Answer

The probability of the coin landing heads exactly 7 times out of 10 flips is approximately 0.246. The conditional probability that exactly 7 out of the 10 flips land on heads, given that the first flip lands on heads, is approximately 0.317.

Step 1: Calculate the probabilities for each coin separately

To find the probability of 7 heads out of 10 for both coins, we will first calculate the probabilities of this event for each coin separately. For Coin 1 (probability of heads = 0.4): P(7 heads out of 10 flips) = \( \binom{10}{7} \times 0.4^7 \times (1 - 0.4)^3 \) For Coin 2 (probability of heads = 0.7): P(7 heads out of 10 flips) = \( \binom{10}{7} \times 0.7^7 \times (1 - 0.7)^3 \) Now, calculate the values for these probabilities.

Step 2: Probability of choosing each coin

We know that either Coin 1 or Coin 2 can be chosen. Since they are randomly chosen, the probability of choosing each coin is 0.5. We will denote these probabilities as P(C1) and P(C2), respectively: P(C1) = 0.5 P(C2) = 0.5

Step 3: Find the total probability

Now, we need to find the total probability of getting 7 heads out of 10 flips. This can be calculated using the total probability theorem: P(7 heads) = P(7 heads|C1) * P(C1) + P(7 heads|C2) * P(C2) Now, plug in the values calculated in steps 1 and 2 to find the total probability.

Step 4: Calculate the probability of 7 heads given the first flip is a head

For the conditional probability, we need to find the probability of exactly 7 heads out of 10 flips, given that the first flip is a head. We can do this using Bayes' theorem: P(7 heads | 1st flip is a head) = [P(1st flip is a head | 7 heads) * P(7 heads)] / P(1st flip is a head) We already know P(7 heads) from Step 3. Therefore, we now need to find P(1st flip is a head | 7 heads) and P(1st flip is a head). For P(1st flip is a head | 7 heads), it can be calculated for each coin separately: P(1st flip is a head | 7 heads, C1) = 1 P(1st flip is a head | 7 heads, C2) = 1 Now, calculate P(1st flip is a head): P(1st flip is a head) = P(1st flip is a head | C1) * P(C1) + P(1st flip is a head | C2) * P(C2) Plug in the values and calculate the conditional probability. This step-by-step solution will give the answers to both (a) and (b) parts of the exercise.

Key Concepts

Understanding Conditional Probability

Conditional probability is crucial when the outcome of an event is influenced by the occurrence of a previous event. In our exercise, we deal with the scenario where prior knowledge of a coin landing on heads affects the probability of getting a specific number of heads in subsequent flips. For example, if we know the first flip is heads, we're left with 9 more flips to reach our total of 7 heads.

This leads us to ask: 'What is the probability of obtaining exactly 6 more heads in the remaining 9 flips?' To calculate this, we use the formula for conditional probability: \[P(A | B) = \frac{P(A \cap B)}{P(B)}\] Here, \( A \) is the event of getting 6 more heads in 9 flips, and \( B \) is the event that the first flip is heads. The numerator is the joint probability of both \( A \) and \( B \) happening, which in this case can be calculated using the binomial distribution for the remaining 9 flips.

When the conditions change (like knowing the outcome of the first flip), we must adjust our calculations to reflect this new information. That's what makes conditional probability so powerful and also a bit tricky—it accounts for changing circumstances to give a more accurate picture of potential outcomes.

Binomial Distribution in Coin Flips

The binomial distribution is a statistical model that's widely used for predicting outcomes in experiments like coin flips, where there are two possible outcomes. It tells us the probability of obtaining a certain number of successes (like heads) in a fixed number of trials, given the probability of success on a single trial.

In our example, the probability of heads on each flip differs for the two coins. To compute the likelihood of flipping heads exactly seven times out of ten, we apply the binomial distribution formula: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\] where \( \binom{n}{k} \) is the binomial coefficient, \( p \) is the probability of success on each trial, \( n \) is the number of trials, and \( k \) is the number of successes. Here, the binomial coefficient computes the number of ways \( k \) successes can occur in \( n \) trials, which is essential when each trial is independent of the others.

We calculate this separately for each coin, as they have different probabilities of landing on heads. Ultimately, the overall probability is a sum of weighted probabilities of each coin, reflecting how the binomial distribution applies to separate and combined events.

Applying Bayes' Theorem to Coin Flips

Bayes' theorem offers a way to revise existing predictions or theories based on new evidence, a process known as Bayesian inference. It's formula is a way to calculate the probability of an event based on prior knowledge of conditions related to the event. The theorem can be stated as: \[ P(A | B) = \frac{P(B | A) \times P(A)}{P(B)} \] In the context of our coin-flipping problem, we use Bayes' theorem to update the likelihood of flipping exactly seven heads given that we know the first flip was heads.

We consider the updated information that the first flip is heads, then calculate how this affects the probability of seven out of ten heads for our two coins—the conditional probability given new evidence. By integrating Bayes' theorem, we are enhancing our probability calculations to be in alignment with the observed outcomes (like the first flip), which speaks to the idea of learning from data and refining our predictions accordingly.