Problem 53 - Basic Statistics for Business and Economics with Student

Question

The probability that a cruise missile hits its target on any particular mission is .80. Four cruise missiles are sent after the same target. What is the probability: a. They all hit the target? b. None hit the target? c. At least one hits the target?

Step-by-Step Solution

Answer

a. 0.4096; b. 0.0016; c. 0.9984.

Step 1: Define the Probability of Hitting the Target

The probability that a single missile hits the target is \( p = 0.80 \). Consequently, the probability that a single missile does not hit the target is \( 1 - p = 0.20 \).

Step 2: Probability that All Hit the Target

To find the probability that all four missiles hit the target, we apply the formula for independent events: \( P( ext{all hit}) = p^4 = 0.80^4 \). Calculating this, \( 0.80^4 = 0.4096 \).

Step 3: Probability that None Hit the Target

For none of the missiles to hit the target, each one must miss. The probability that none hit is given by \( P( ext{none hit}) = (1-p)^4 = 0.20^4 \). Calculating this, \( 0.20^4 = 0.0016 \).

Step 4: Probability that At Least One Hits the Target

The probability that at least one missile hits is the complement of none hitting. It can be calculated as: \( P( ext{at least one hit}) = 1 - P( ext{none hit}) \). Thus, \( 1 - 0.0016 = 0.9984 \).

Key Concepts

Independent Events

In probability theory, independent events are events whose outcomes do not affect each other. For example, if you have a coin flip and a die roll, the result of the coin flip does not change the likelihood of any outcome with the die roll. Each event's result stands alone.

In our example of cruise missiles, each missile's success or failure in hitting the target is considered independent. This means the outcome of one missile does not impact the others. We can calculate the probability of all missiles hitting or missing the target by multiplying the probabilities together for each individual event.

• For all hitting the target: Multiply the success probability of each missile (0.80) together four times: \(0.80 \times 0.80 \times 0.80 \times 0.80 = 0.80^4 \).
• For all missing the target: Do the same with the probability of failure (0.20): \(0.20 \times 0.20 \times 0.20 \times 0.20 = 0.20^4 \).

These calculations use the principle of independent events to find the likelihood of multiple scenarios occurring at once.

Complement Rule

The complement rule in probability is a powerful concept that simplifies the process of finding probabilities. Rather than directly calculating the probability of an event happening, you find the probability of it not happening and subtract it from one.

In the exercise, to find the probability that at least one missile hits the target, you can think backwards. Instead of calculating directly, find the probability that none of the missiles hit, and subtract this value from one. This uses the complement rule.

• The probability that none hit is \(0.20^4\), since each one misses independently.
• To find at least one hit: \(1 - 0.20^4 = 1 - 0.0016 = 0.9984\).

This approach leverages how much easier it is to deal with 'none' or 'all' scenarios, allowing us to find the complementary outcome in a straightforward way.

Binomial Probability

Binomial probability takes into account multiple trials of an event with two possible outcomes, like success and failure. In scenarios like our cruise missiles, each trial (each missile) is considered identically independent.

The binomial probability formula is used to calculate the probability of achieving a certain number of successes in a set number of trials. It is generally represented as:

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

where

• \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\), indicating the number of ways to choose \(k\) successes from \(n\) trials.
• \(p\) is the probability of a single success.
• \(n\) is the total number of trials.
• \(k\) is the number of successes you're calculating for.

Even though we didn't use the full binomial formula in the step-by-step solution, the calculations conceptually align with binomial scenarios where you'd usually account for different outcomes and some complexities of multiple outcomes in given trials.