Question

1. zach’s teammate, taylor, is the quarterback for the football team. she can complete 65% of her passes. for the following table, fill in the missing pieces.

situation	probability notation	formula	value
what is the probability that it takes taylor	$p(x < 3)$
		$(0.35)^4(0.65)$
		$1 - (p(x = 1) + p(x = 2) + p(x = 3) + p(x = 4) + p(x = 5))$

Explanation:

Step1: Determine the formula for \(P(X = k)\)

For a geometric distribution, the formula is \(P(X=k)=(1 - p)^{k}p\), where \(p = 0.65\) (probability of success - a completed pass) and \(1-p=0.35\) (probability of failure - an incomplete pass). For \(P(X = 3)\), substitute \(k = 3\) into the formula: \(P(X = 3)=(0.35)^{3}(0.65)\).

Step2: Calculate \(P(X = 3)\)

\((0.35)^{3}(0.65)=0.35\times0.35\times0.35\times0.65 = 0.02756875\approx0.0276\)

Step3: For \(P(X\lt3)\)

\(P(X\lt3)=P(X = 0)+P(X=1)+P(X = 2)\)
Using the geometric formula \(P(X=k)=(1 - p)^{k}p\):
\(P(X = 0)=(0.35)^{0}(0.65)=0.65\)
\(P(X = 1)=(0.35)^{1}(0.65)=0.2275\)
\(P(X = 2)=(0.35)^{2}(0.65)=0.35\times0.35\times0.65 = 0.079625\)
\(P(X\lt3)=0.65 + 0.2275+0.079625=0.957125\approx0.9571\)

Step4: For the formula \((0.35)^{4}(0.65)\)

The situation is "What is the probability that it takes Taylor 4 incomplete passes before she has a completion?" (since \(k = 4\) in \(P(X=k)=(1 - p)^{k}p\))

Step5: For the formula \(1-(P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)+P(X = 5))\)

The situation is "What is the probability that it takes Taylor more than 5 incomplete passes before she has a completion?" (using the complement rule \(P(X\gt n)=1 - P(X\leq n)\))

Answer:

Situation	Probability Notation	Formula	Value
What is the probability that it takes Taylor less than 3 incomplete passes before she has a completion?	\(P(X\lt3)\)	\(P(X = 0)+P(X = 1)+P(X = 2)\)	\(0.9571\)
What is the probability that it takes Taylor 4 incomplete passes before she has a completion?	\(P(X = 4)\)	\((0.35)^{4}(0.65)\)	\(0.009649\) (calculated as \(0.35\times0.35\times0.35\times0.35\times0.65\))
What is the probability that it takes Taylor more than 5 incomplete passes before she has a completion?	\(P(X\gt5)\)	\(1-(P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)+P(X = 5))\)	\(1-(0.2275 + 0.079625+0.02756875+0.009649+0.003377125)=0.65227\)