Question

The National Hockey League (NHL) has \(80 \%\) of its players born outside the United States, and of those born outside the United States, \(50 \%\) are born in Canada. \(^{2}\) Suppose that \(n=12\) NHL players were selected at random. Let \(x\) be the number of players in the sample who were born outside of the United States so that \(p=.8\). Find the following probabilities: a. At least five or more of the sampled players were born outside the United States. b. Exactly seven of the players were born outside the United States. c. Fewer than six were born outside the United States.

Short answer

a. At least five or more players were born outside the United States. b. Exactly seven of the players were born outside the United States. c. Fewer than six were born outside the United States.

Step-by-step solution

(a. At least five or more players were born outside the United States.)

To find the probability of at least 5 players born outside the United States, we need to find the sum of probabilities for 5, 6, 7,..., up to 12 players born outside the United States. \(P(x \geq 5) = P(x=5) + P(x=6) + \cdots + P(x=12)\) Now, we will use the binomial probability formula to find the probability of each individual case and then sum those probabilities. \(P(x \geq 5) = \sum_{x=5}^{12} \binom{12}{x} (0.8)^x (0.2)^{12-x}\) Computing the probabilities and summing them will give us the value for P(x ≥ 5).

(b. Exactly seven of the players were born outside the United States.)

For this part of the problem, we will use the binomial probability formula to find the probability of exactly 7 players being born outside the United States. \(P(x=7) = \binom{12}{7} (0.8)^7 (0.2)^{12-7}\) Calculating the probability will give us the value for P(x=7).

(c. Fewer than six were born outside the United States.)

To find the probability of fewer than six players born outside the United States, we need to find the sum of probabilities for 0, 1, 2, 3, 4, and 5 players born outside the United States. \(P(x < 6) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5)\) Now, we will use the binomial probability formula to find the probability of each individual case and then sum those probabilities. \(P(x < 6) = \sum_{x=0}^{5} \binom{12}{x} (0.8)^x (0.2)^{12-x}\) Computing the probabilities and summing them will give us the value for P(x < 6).

Key concepts

Probability Distribution

A probability distribution is a function that describes the likelihood of various outcomes in a random experiment. In the context of the binomial distribution, which is used here, it helps to determine the probability of a certain number of successes in a series of independent experiments.

For our exercise involving NHL players, each player can be seen as a trial or experiment. The outcome is whether each player is born outside the United States or not. When you're analyzing probability distribution, you're examining all possible outcomes and how likely each one is.

Think of it as a way of quantifying your expectations in a probabilistic form. For example, there are different probabilities for having 0 to 12 players born outside the United States, which together form the probability distribution of the outcomes.

Binomial Formula

The binomial formula is key when calculating probabilities for events with two outcomes: success and failure. It determines the probability of getting exactly a certain number of successes in a fixed number of trials. This is essential for exercises like ours.

The formula is:

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

Where:

• \(n\) is the number of trials (players sampled)
• \(x\) is the number of successful outcomes (players born outside the U.S.)
• \(p\) is the probability of success on a single trial (0.8 in this case)
• \(1-p\) is the probability of failure (0.2 here)

This formula helps us calculate each specific probability outcome in our scenarios (e.g., exactly 7 players or less than 6 born outside the U.S.).

Sample Size

Sample size refers to the number of observations or trials being considered in a statistical analysis. In our exercise, the sample size is 12, meaning that 12 NHL players are randomly selected to evaluate as part of this probability question.

Sample size is important because it influences the precision of our probability estimates. A larger sample size tends to give a more accurate reflection of the true nature of the population, assuming the sample is representative.

For instance, analyzing 12 players gives us a fairly substantial sample size to draw meaningful probabilistic conclusions regarding how many players may or may not have been born inside the United States.

Event Probability

Event probability refers to the likelihood of a specific outcome occurring during an experiment. In the context of our binomial probability scenario, event probability is about figuring out the likelihood of a certain number of players being born outside the United States.

When dealing with event probability, we're interested in outcomes like:

• At least 5 players born outside the U.S. (\(P(x \geq 5)\))
• Exactly 7 players born outside the U.S. (\(P(x = 7)\))
• Less than 6 players born outside the U.S. (\(P(x < 6)\))

Each of these outcomes has a calculated probability using the binomial formula, providing insights into how frequently each event will happen in this scenario.