Problem 42 - Basic Statistics for Business & Economics

Question

The Bank of Hawaii reports that \(7 \%\) of its credit card holders will default at some time in their life. The Hilo branch just mailed out 12 new cards today. a. How many of these new cardholders would you expect to default? What is the standard deviation? b. What is the likelihood that none of the cardholders will default? c. What is the likelihood at least one will default?

Step-by-Step Solution

Answer

a. Expect 0.84 defaults, standard deviation 0.8676. b. Likelihood none default: 0.4783. c. Likelihood at least one defaults: 0.5217.

Step 1: Determine Expected Number of Defaults

To find the expected number of cardholders who will default, use the formula for expected value in a binomial distribution: \[ E(X) = n \cdot p \]where \(n\) is the number of trials (12 cardholders), and \(p\) is the probability of default (0.07). Plugging in the values:\[ E(X) = 12 \cdot 0.07 = 0.84 \]Thus, we expect 0.84 cardholders to default.

Step 2: Calculate Standard Deviation

The standard deviation for a binomial distribution is found using the formula:\[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \]where \(n = 12\) and \(p = 0.07\). Calculating this gives:\[ \sigma = \sqrt{12 \cdot 0.07 \cdot 0.93} \approx 0.8676 \] So, the standard deviation is approximately 0.8676.

Step 3: Calculate Probability of No Defaults

The probability that none of the cardholders will default is given by the probability mass function for a binomial random variable:\[ P(X = 0) = \binom{n}{0} p^0 (1-p)^{n} \]Substitute \(n = 12\) and \(p = 0.07\):\[ P(X = 0) = (1 - 0.07)^{12} = 0.93^{12} \approx 0.4783 \]Thus, the likelihood that none of the cardholders will default is approximately 0.4783.

Step 4: Calculate Probability of At Least One Default

The probability of at least one default is the complement of the probability of no defaults. Therefore, use:\[ P(X \geq 1) = 1 - P(X = 0) \]From the previous step:\[ P(X \geq 1) = 1 - 0.4783 = 0.5217 \]The likelihood that at least one cardholder will default is approximately 0.5217.

Key Concepts

Expected Value

In a binomial distribution, the expected value is a measure of the center of the probability distribution. It's basically the average number of successes you'd expect to see if you repeated the experiment many times. In the context of the problem provided, the expected value tells us how many cardholders from the Hilo branch might default on their credit cards based on historical data.
The formula to calculate the expected value for a binomial distribution is \( E(X) = n \cdot p \), where \( n \) represents the number of trials, or in this case, the number of new credit cardholders (12). The \( p \) is the probability of success, which here is the 7% default rate (0.07) reported by the Bank of Hawaii.

• Expected Value (\( E(X) \)): It is calculated by multiplying the total number of cardholders by the probability of default. Thus, \( E(X) = 12 \cdot 0.07 = 0.84 \).

This means, on average, you can expect about 0.84 of the new cardholders to default, which can be interpreted as less than one cardholder on average.

Standard Deviation

The standard deviation in a binomial distribution provides insight into the variability around the expected value. It's a measure of how much we might expect the actual number of defaults to "deviate" or differ from the expected value of 0.84 over multiple trials.
The formula for the standard deviation of a binomial distribution is \( \sigma = \sqrt{n \cdot p \cdot (1-p)} \). Here, \( n \) is the number of trials (12 cardholders), \( p \) is the probability of a cardholder defaulting (0.07), and \( 1-p \) represents the probability of a cardholder not defaulting (0.93).

• Standard Deviation (\( \sigma \)): Applying the formula, we find \( \sigma = \sqrt{12 \cdot 0.07 \cdot 0.93} \approx 0.8676 \).

This tells us that the number of cardholders who might default is likely to vary by approximately 0.8676 cardholders from the expected value of 0.84.

Probability Mass Function

The Probability Mass Function (PMF) is a function that gives us the probability of a discrete random variable being exactly equal to some value. For a binomial distribution, it helps us calculate the probability of a specific number of successes, which, in this case, is the number of defaults.
To determine the probability that none of the cardholders defaults, we use the PMF formula for a binomial random variable:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
For no defaults \((k = 0)\), the calculation simplifies, as the term \( p^0 \) equals 1, and you only need \( (1-p)^n \). With 12 cardholders and a 0.07 default probability:

• Probability of No Defaults: \( P(X = 0) = 0.93^{12} \approx 0.4783 \).

This probability of about 0.4783 is the chance that none of the cardholders will default.
To find the probability that at least one person defaults, note that it is the opposite of none defaulting. So, you subtract the probability of none defaulting from 1:

• Probability of At Least One Default: \( P(X \geq 1) = 1 - 0.4783 = 0.5217 \).

Therefore, there's roughly a 52.17% chance that at least one cardholder will default.