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The binomial distribution is a discrete probability distribution. It models the number of successes in a fixed number of independent trials of a binary experiment. A binary experiment is one in which there are two possible outcomes: success or failure. In this context, we're examining whether adults get enough leisure time exercise or not. Each adult considered is an individual trial. For a binomial distribution, we need:

• A fixed number of trials, represented as \( n \). In our exercise scenario, \( n = 3 \) since we are examining the exercise habits of 3 adults.
• A constant probability of success on each trial, represented as \( p \). Here, \( p = 0.27 \), reflecting that 27% of U.S. adults get enough leisure time exercise.

The outcomes of the trials are independent, meaning the outcome of one trial does not affect the others.The binomial probability formula is: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.

Probability in a binomial experiment involves determining the likelihood of a certain number of successes. Using the binomial formula, we calculate these probabilities for specific scenarios. Let's look at two cases:For part (a), we calculated the probability that all 3 adults get enough exercise, which translates to 3 successes. Plugging into the formula: \[P(X=3) = \binom{3}{3} (0.27)^3 (0.73)^0\]This calculation involves:

• Determining \( \binom{3}{3} \), which is 1, as there is only one way to have all 3 successes.
• Raising 0.27 (probability of success) to the 3rd power, resulting in \( (0.27)^3 \).

So, the probability that all adults get enough exercise is approximately 0.0197.For part (b), the question asked for the probability that at least 1 adult gets enough exercise. Here, it is efficient to apply the complement rule for a swifter calculation.

The complement rule is a handy technique in probability calculations, particularly when dealing with 'at least one' scenarios. Rather than calculating the probabilities of 1, 2, and 3 successes separately, which can be cumbersome, we focus on its complement, which is usually much easier.In our exercise problem, finding the probability that at least 1 adult gets enough exercise is simplified by first calculating the probability of the complementary scenario: no adult getting enough exercise (0 successes):Using the formula from the previous section:\[P(X=0) = \binom{3}{0} (0.27)^0 (0.73)^3\]This breaks down to:

• \( \binom{3}{0} = 1 \), since there is only one way to have no successes.
• Raising 0.73 (probability of no success) to the power 3, yielding \( (0.73)^3 \).

The probability that no adult gets enough exercise is approximately 0.3890.Using the complement rule, we subtract this from 1 to find the probability that at least 1 adult gets enough exercise:\[P(X \geq 1) = 1 - 0.3890 = 0.6110\]So, the probability that at least 1 of the 3 adults gets enough exercise is about 0.6110.