Question

information on the weight and years of experience for a football team were obtained. the given contingency table provides a cross - classification of a player on the football team is selected at random. complete parts (a) and (b) below. click to view the contingency table.
a. the probability that the player selected is a rookie = \square (round to three decimal places as needed.)
b. the probability that the player selected is a rookie, given that he weighs under 200 pounds = \square (round to three decimal places as needed.)
data table
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$$\begin{tabular}{|c|c|c|c|c|c|} \\hline \\multirow{5}{*}{weight (lb)} & & \\multicolumn{4}{c|}{years of experience} & \\multirow{2}{*}{total} \\\\ \\cline{3 - 6} & & rookie $y_1$ & 1 - 5 $y_2$ & 6 - 10 $y_3$ & over 10 $y_4$ & \\\\ \\hline & under 200 $w_1$ & 5 & 5 & 1 & 0 & 11 \\\\ \\hline & 200 - 300 $w_2$ & 8 & 10 & 16 & 8 & 42 \\\\ \\hline & over 300 $w_3$ & 0 & 8 & 7 & 0 & 15 \\\\ \\hline & total & 13 & 23 & 24 & 8 & 68 \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Calculate the probability for part (a)

The formula for probability is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For part (a), the number of rookies (\(Y_1\)) is \(13\) and the total number of players is \(68\). So \(P(\text{rookie})=\frac{13}{68}\).

$$ \frac{13}{68}\approx 0.191 $$

Step2: Calculate the probability for part (b)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is the event of being a rookie and \(B\) is the event of weighing under \(200\) pounds.
The number of players who are rookies and weigh under \(200\) pounds (\(Y_1\cap W_1\)) is \(5\), and the number of players who weigh under \(200\) pounds (\(W_1\)) is \(11\). So \(P(\text{rookie}|\text{under }200)=\frac{5}{11}\).

$$ \frac{5}{11}\approx 0.455 $$

Answer:

a. \(0.191\)
b. \(0.455\)