Question

Cancelled flights. Mary is deciding whether to book the cheaper flight home from college after her final exams, but she's unsure when her last exam will be. She thinks there is only a \(20 \%\) chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is and she has to cancel the flight, she will lose \(\$ 150\). If she can take the cheaper flight, she will save \(\$ 100\). a) If she books the cheaper flight, what can she expect to gain, on average? b) What is the standard deviation?

Short answer

Mary's expected gain is $50, with a standard deviation of $100.

Step-by-step solution

Define Expected Gain

To find the expected gain, we'll multiply each outcome's gain by its probability and sum these values. The gain if she can take the flight is \(+100\) dollars, and the probability is \(80\%\). The gain if she cannot take the flight is \(-150\) dollars, and the probability is \(20\%\). We'll use the expected value formula: \[ E(X) = \sum (x_i \cdot P(x_i)) \] where \(x_i\) is the gain and \(P(x_i)\) is the probability.

Calculate Expected Gain

Using the formula, compute: \[ E(X) = (+100 \times 0.80) + (-150 \times 0.20) \] This equals: \[ E(X) = 80 - 30 = 50 \] Mary's expected gain is \(\$50\).

Define Variance Formula

The variance measures the spread of possible outcomes. We calculate it using the formula: \[ \text{Var}(X) = \sum ( (x_i - E(X))^2 \cdot P(x_i)) \] where \(x_i\) is each outcome, and \(E(X)\) is the expected gain calculated previously.

Calculate Variance

Substitute the outcomes and probabilities into the variance formula. Since \(E(X) = 50\), compute: \[ \text{Var}(X) = ((100 - 50)^2 \times 0.80) + ((-150 - 50)^2 \times 0.20) \] This equals: \[ \text{Var}(X) = (50^2 \times 0.80) + (200^2 \times 0.20) \] \[ \text{Var}(X) = (2500 \times 0.80) + (40000 \times 0.20) \] \[ \text{Var}(X) = 2000 + 8000 = 10000 \]

Calculate Standard Deviation

The standard deviation is the square root of the variance: \[ \text{SD}(X) = \sqrt{\text{Var}(X)} = \sqrt{10000} \] This equals \(100\). The standard deviation of her gain is \(\$100\).

Key concepts

Probability Distribution

Probability distribution is a key concept in statistics that helps us understand how the likelihood of different outcomes is spread. In Mary's case, we have two outcomes:

• She takes the cheaper flight and saves \( \\(100 \).
• She misses the flight, hence cancels it and loses \( \\)150 \).

For each of these outcomes, we attach a probability. The main goal is to map each potential gain or loss to a probability, essentially creating a probability distribution.
For Mary:

• The probability of saving \( \\(100 \) by taking the flight is \(80\%\) or \(0.8\).
• The probability of losing \( \\)150 \) by not taking the flight is \(20\%\) or \(0.2\).

These probabilities must add up to \(1\), ensuring that all possible outcomes are accounted for. By understanding these distributions, we can make informed predictions about future events, like Mary's expected monetary gain.

Variance

Variance is a statistical measure that tells us how much the outcomes in the probability distribution deviate from the expected value. It essentially quantifies the degree of spread or dispersion we can observe around the mean.
For Mary, we calculated the expected gain (mean), which is \(\\(50\). Now, to find the variance, we evaluate how each outcome differs from these \(\\)50\) and square that difference. Squaring makes all deviations positive and gives more weight to larger deviations:

• For saving \( \\(100 \), the deviation is \(100 - 50 = \\)50\). The square is \(50^2 = 2500\).
• For losing \( \\(150 \), the deviation is \( -150 - 50 = -\\)200\). The square is \(200^2 = 40000\).

Each squared deviation is weighted by its probability before being summed to find the overall spread. The resulting variance for Mary's situation is \(10000\).
This tells us how much risk or variability is present in her potential financial outcome.

Standard Deviation

The standard deviation is an essential concept that builds on variance. While variance provides a measure in squared units, the standard deviation brings it back to the original units, such as dollars, making it more interpretable.
Calculating the standard deviation involves taking the square root of the variance. For Mary's scenario, where the variance is \(10000\), the standard deviation is:

• \(\text{SD}(X) = \sqrt{10000} = 100\)

A standard deviation of \( \\(100 \) means that the typical deviation from Mary's expected gain of \( \\)50 \) is \( \$100 \). This measure highlights how much uncertainty surrounds her expected outcome and helps her understand potential risks or fluctuations in the gain or loss associated with her flight booking.