Question

julie has a stack of 40 cards. each card has a color and a number. the colors and numbers of each card in the deck are shown in the table.\
colors\tnumbers\
red\\t1 to 10\
green\\t1 to 10\
blue\\t1 to 10\
purple\\t1 to 10\
julie draws a random card. let a be the event that the card has the number 8. let b be the event that the card is blue.\
$p(a)=\frac{1}{10}$, $p(b)=\frac{1}{4}$, and $p(a \text{ and } b)=\frac{1}{40}$\
move the options to the spaces to complete the sentence.\
the events a and b are \underline{\hspace{1cm}} because $p(a \text{ and } b)$ \underline{\hspace{1cm}} $p(a)$ \underline{\hspace{1cm}} $p(b)$.\
options: dependent, independent, $=$, $\
eq$, $+$, $\times$

Explanation:

Step1: Recall independent events formula

For two events \( A \) and \( B \), if they are independent, then \( P(A \text{ and } B) = P(A) \times P(B) \).
Calculate \( P(A) \times P(B) \): \( P(A)=\frac{1}{10} \), \( P(B)=\frac{1}{4} \), so \( P(A) \times P(B)=\frac{1}{10} \times \frac{1}{4}=\frac{1}{40} \).

Step2: Compare with \( P(A \text{ and } B) \)

We know \( P(A \text{ and } B)=\frac{1}{40} \), which is equal to \( P(A) \times P(B) \). So events \( A \) and \( B \) are independent because \( P(A \text{ and } B) = P(A) \times P(B) \).

Answer:

The events \( A \) and \( B \) are \(\boldsymbol{\text{independent}}\) because \( P(A \text{ and } B) \boldsymbol{=} P(A) \boldsymbol{\times} P(B) \).