Problem 34 - Calculus for Biology and Medicine

Question

Toss a coin with probability of heads \(0.3\) five times. Let \(X\) be the number of tails. Find (a) \(P(X=2)\). (b) \(P(X \geq 1)\).

Step-by-Step Solution

Answer

(a) 0.1323. (b) 0.99757.

Step 1: Identify the Probability of Tails

Since the probability of getting heads is 0.3, the probability of getting tails must be the complement of this. Thus, we calculate the probability of tails as: \[ P(Tails) = 1 - 0.3 = 0.7 \]

Step 2: Define the Binomial Distribution

This is a binomial distribution problem because we are dealing with a fixed number of independent trials, each trial has two possible outcomes (heads or tails), and the probability of success (getting tails) is the same for each trial. Here, the number of trials \(n = 5\) and \(p = 0.7\).

Step 3: Calculate the Probability for Part (a)

To find \(P(X = 2)\), where \(X\) is the number of tails, we use the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substitute the given values:\[ P(X = 2) = \binom{5}{2} (0.7)^2 (0.3)^{5-2} \]Calculate further:\[ \binom{5}{2} = 10,\ (0.7)^2 = 0.49,\ (0.3)^3 = 0.027 \]Thus,\[ P(X = 2) = 10 \times 0.49 \times 0.027 = 0.1323 \]

Step 4: Calculate the Probability for Zero Tails

For part (b), find the probability of zero tails to later calculate \(P(X \geq 1)\). Use:\[ P(X = 0) = \binom{5}{0} (0.7)^0 (0.3)^5 \]Since \( \binom{5}{0} = 1\), \((0.7)^0 = 1\), and \((0.3)^5 \approx 0.00243\), we get:\[ P(X = 0) = 1 \times 1 \times 0.00243 = 0.00243 \]

Step 5: Find the Probability of At Least One Tail

Now use the complementary rule: \[ P(X \geq 1) = 1 - P(X = 0) \]Substitute the computed probabilities:\[ P(X \geq 1) = 1 - 0.00243 = 0.99757 \]

Key Concepts

Probability

Probability is the measure of how likely an event is to occur. Imagine flipping a coin.
Each flip has a chance to result in heads or tails. If a coin is fair, the probability of each result, heads or tails, is 0.5.
However, in our problem, the probability of getting heads is 0.3. This means the coin is not fair since each outcome doesn't have equal chances.
By understanding probability, we can calculate the likelihood of different outcomes, such as getting a certain number of tails in several coin tosses.
In our exercise, this understanding helps us figure out what happens over five flips.
This involves finding the chances of getting two tails or at least one tail.

Independent Trials

When we refer to independent trials, it means that the outcome of one trial doesn't affect the next.
For example, flipping a coin is an independent event. What happens in one flip doesn't change the probability of what will happen in the next flip.
This concept is crucial when dealing with binomial distribution problems, like ours.
Every flip of the coin, in our question, is an independent trial.
Even if you get tails in one flip, it does not change the 0.7 probability of getting tails in the next flip. Understanding independent trials lets us treat each coin flip as a fresh event,
unaffected by the previous outcomes, simplifying the way we calculate probabilities in multiple attempts.

Complementary Rule

The complementary rule is a handy tool in probability, helping us find probabilities by looking at what does not happen.
It states that the probability of an event happening plus the probability of it not happening equals 1.
In our exercise, finding the probability of getting no tails helps us figure out the probability of getting at least 1 tail.
It uses this complementary principle, where:

• Probability of at least 1 tail = 1 - Probability of no tails.

This trick is great for simplifying problems, especially when it's easier to calculate the opposite of what we want.

Binomial Probability Formula

The binomial probability formula helps us determine the probability of getting a specific number of 'successes' in a series of independent trials.
This is crucial for scenarios like ours, where we want to know the precise chances of getting 2 tails in 5 tosses.
The formula looks like this:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:

• \( \binom{n}{k} \) is the combination of choosing \( k \) successes in \( n \) trials.
• \( p \) is the probability of success (getting tails), \( 0.7 \) in our case.
• \( (1-p) \) is the probability of failure (getting heads).

By substituting these values into the formula, we can find the exact probability of events,
such as getting 2 tails or determining outcomes like in part (a) and (b) of our exercise.