Part 2 of 4 (b) What is the probability that more than 3 weigh more than 20 pounds? The probability that more than 3 weigh more than 20 pounds is $$ \square $$.

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Let $$X$$ be the number of items that weigh more than 20 pounds. If we assume that $$X$$ follows a binomial distribution with parameters $$n$$ (the total number of items weighed) and $$p$$ (the probability that a single item weighs more than 20 pounds), then

$$
X \sim \operatorname{Binomial}(n,p).
$$

We are asked to find the probability that more than 3 items have a weight greater than 20 pounds, which is

$$
P(X > 3).
$$

Since the total probability is 1, we can write this probability as the complement of the event that 3 or fewer items weigh more than 20 pounds:

$$
P(X > 3) = 1 - P(X \le 3).
$$

The probability of 3 or fewer items weighing more than 20 pounds is given by the cumulative probability

$$
P(X \le 3) = \sum_{i=0}^{3} \binom{n}{i} p^i (1-p)^{\,n-i}.
$$

Thus, the probability that more than 3 items weigh more than 20 pounds is

$$
P(X > 3) = 1 - \sum_{i=0}^{3} \binom{n}{i} p^i (1-p)^{\,n-i}.
$$

The final answer is

$$
\boxed{1 - \sum_{i=0}^{3} \binom{n}{i} p^i (1-p)^{\,n-i}}.
$$
The probability that more than 3 items weigh more than 20 pounds is $$1 - \sum_{i=0}^{3} \binom{n}{i} p^i (1-p)^{n-i}$$.

Part 2 of 4 (b) What is the probability that more than 3 weigh more than 20 pounds? The probability that more than 3 weigh more than 20 pounds is \( \square \).

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