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The binomial probability formula is used to determine the probability of obtaining exactly $$k$$ successes in a sequence of $$n$$ independent trials, each with two outcome categories (success and failure) and the same probability of success. The formula is expressed as $$\binom{n}{k} p^k (1-p)^{n-k}$$, where $$p$$ represents the probability of success on a single trial, and the binomial coefficient $$\binom{n}{k}$$ calculates the number of ways to choose $$k$$ successes from $$n$$ trials.

Binomial Probability Formula

To calculate the probability of getting exactly one head in three independent coin flips, the binomial formula can be applied. In this scenario, the number of trials ($$n$$) is 3, the desired number of successes ($$k$$) is 1, and the probability of a single success (heads, $$p$$) is 0.5. Substituting these values into the formula, $$\binom{3}{1}(0.5)^1(0.5)^{3-1}$$, results in a calculation of $$ 3 \cdot 0.5 \cdot 0.25 $$, which equals 0.375.

Applying the Binomial Formula for One Head in Three Flips

A basketball player has a consistent success rate of 80% on free throws. Assuming each shot is an independent event, first, explain why this scenario is suitable for a probability model that deals with a fixed number of trials where each trial has only two outcomes. Second, calculate the exact probability that this player will make exactly 4 out of their next 5 free-throw attempts. Show your setup.

Free-Throw Probability Analysis

Based on the provided scenario, evaluate whether the analyst's chosen formula is appropriate. Justify your conclusion by analyzing the characteristics of the card-drawing experiment against the requirements of the formula.

Evaluating a Statistical Model for Card Draws

A factory manufactures light bulbs with a 5% defect rate, where the status of each bulb is independent of the others. A quality control inspector randomly selects a sample of 20 bulbs. Which of the following expressions correctly represents the probability that exactly 3 of the sampled bulbs are defective?

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