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) \(P(X=2) = 0.1323\). (b) \(P(X \geq 1) = 0.99757\).

Step 1: Define the probability of tails

Since the probability of getting heads is 0.3, the probability of getting tails is the complement of this. Calculate the probability of getting tails: \( P(T) = 1 - P(H) = 1 - 0.3 = 0.7 \).

Step 2: Model the problem as a binomial distribution

The random variable \(X\), which represents the number of tails, follows a binomial distribution with parameters \(n = 5\) (number of tosses) and \(p = 0.7\) (probability of tails). We denote this as \(X \sim \text{Binomial}(5, 0.7)\).

Step 3: Calculate \(P(X=2)\) using the binomial formula

The probability \(P(X = k)\) in a binomial distribution is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substituting \( n = 5 \), \( k = 2 \), and \( p = 0.7 \): \[ P(X=2) = \binom{5}{2} (0.7)^2 (0.3)^3 \].

Step 4: Calculate \(\binom{5}{2}\)

Calculate the combination \(\binom{5}{2}\), which represents the number of ways to choose 2 tails out of 5 tosses: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \].

Step 5: Complete the calculation of \(P(X=2)\)

Now, substitute the values into the formula: \[ P(X=2) = 10 \times (0.7)^2 \times (0.3)^3 = 10 \times 0.49 \times 0.027 = 0.1323 \].

Step 6: Calculate \(P(X \geq 1)\)

First, recognize that \(P(X \geq 1) = 1 - P(X = 0)\). So we need to calculate \(P(X=0)\) using the binomial formula: \[ P(X=0) = \binom{5}{0} (0.7)^0 (0.3)^5 = 1 \times 1 \times 0.00243 = 0.00243 \].

Step 7: Find \(P(X \geq 1)\)

Subtract \(P(X=0)\) from 1 to find \(P(X \geq 1)\): \[ P(X \geq 1) = 1 - P(X=0) = 1 - 0.00243 = 0.99757 \].

Key Concepts

Probability of Tails

When tossing a coin, knowing the probability of tails helps us calculate the chances of certain outcomes. The probability of getting tails is simply the complement of getting heads. This means if you know the probability of heads, say 0.3, then the probability of tails will be what remains in the total probability. The total must always equal 1, as all possible outcomes together would summarize the whole certainty of an event.

So for our coin toss example, if the probability of getting a head is 0.3, the probability of getting a tail becomes:

• Subtract the probability of heads from 1
• 1 - 0.3 = 0.7

This tells us that there is a 70% chance of getting tails with each coin toss. Understanding this concept is crucial when using the binomial distribution, as it determines how likely tails will appear over several coin tosses.

Combination Formula

The combination formula \( \binom{n}{k} \) is key in calculating probabilities very efficiently, especially useful in scenarios such as our coin tossing example. This formula helps you find how many ways you can choose \( k \) successes (like getting tails) from \( n \) trials (coin tosses).

The formula is given by:

• \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Here, \( n! \) means "n factorial," which is the product of all positive integers up to \( n \).

Let's take an example. If you want to know how many ways you can get 2 tails in 5 tosses, you calculate \( \binom{5}{2} \):

• Calculate \(5! = 5 \times 4 \times 3 \times 2 \times 1\)
• Calculate the parts for the denominator: \(2!(5-2)! = 2 \times 1 \times 3 \times 2 \times 1\)
• Insert into the formula: \[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

This tells us there are 10 different ways to achieve exactly 2 tails in 5 tosses.

Binomial Probability Formula

The binomial probability formula is essential when you need to determine the likelihood of getting a specific number of successes in a sequence of independent trials. This is typically used in situations where each trial can result in one of two possible outcomes, like heads or tails in a coin toss.

The formula is:

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

Where:

• \( P(X = k) \) is the probability of getting \( k \) successes
• \( n \) is the total number of trials
• \( p \) is the probability of success on an individual trial
• \((1-p)\) is the probability of failure on an individual trial

In our scenario of tossing a coin 5 times, if you want to calculate the probability of getting exactly 2 tails, you substitute in the values:

• Use \( n = 5 \), \( k = 2 \), and \( p = 0.7 \) (tails probability)
• \( P(X=2) = \binom{5}{2} (0.7)^2 (0.3)^3 \)
• \( = 10 \times 0.49 \times 0.027 \)
• \( = 0.1323 \)

The calculated result, 0.1323, means there is a 13.23% chance of getting exactly 2 tails when flipping the coin five times.