Question

aditi downloads ten paid apps and sixteen free apps on her tablet. fourteen of them are game apps, and she paid for five of the game apps. complete the statements to determine if the events “paid” and “game” are independent. p(paid) = p(paid | game) = the events “paid” and “game” are independent not independent

Explanation:

Step1: Calculate \( P(\text{paid}) \)

Total number of apps \(=10 + 16=26\).
Number of paid apps \(=10\).
By the formula \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\), \(P(\text{paid})=\frac{10}{26}=\frac{5}{13}\).

Step2: Calculate \( P(\text{paid}|\text{game}) \)

Number of game apps \(n(\text{game}) = 14\).
Number of paid - game apps \(n(\text{paid and game})=5\).
By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}\), \(P(\text{paid}|\text{game})=\frac{5}{14}\).

Step3: Check for independence

Two events \(A\) (paid) and \(B\) (game) are independent if \(P(A)=P(A|B)\).
Since \(\frac{5}{13}\approx0.385\) and \(\frac{5}{14}\approx0.357\), \(\frac{5}{13}
eq\frac{5}{14}\).

Answer:

\(P(\text{paid})=\frac{5}{13}\), \(P(\text{paid}|\text{game})=\frac{5}{14}\), The events “paid” and “game” are not independent.