Question

if we draw a card from those above, whats the probability that it has rank 5? (enter as a decimal.) .2
which of the options represents the likelihood of drawing a card that is rank 5, given that its gray. choose your answer...
if we draw a card from those above, and notice only that its gray, whats the probability that it has rank 5? (enter as decimal.) type your answer...
are events \gray\ and
ank 5\ independent? choose your answer...

Explanation:

Step1: Count total number of cards

There are 4 rows and 5 columns, so total number of cards $n = 4\times5=20$.

Step2: Count number of cards with rank 5

There are 4 cards with rank 5 (X5, Y5, Z5, W5). So $n_{5}=4$.

Step3: Calculate probability of drawing a card with rank 5

The probability $P(5)=\frac{n_{5}}{n}=\frac{4}{20} = 0.2$.

Step4: Count number of gray - colored cards

There are 10 gray - colored cards. So $n_{gray}=10$.

Step5: Count number of gray - colored cards with rank 5

There are 2 gray - colored cards with rank 5 (Z5, W5). So $n_{gray\&5}=2$.

Step6: Calculate conditional probability

The conditional probability $P(5|gray)=\frac{n_{gray\&5}}{n_{gray}}=\frac{2}{10}=0.2$.

Step7: Check for independence

Two events $A$ (card is gray) and $B$ (card has rank 5) are independent if $P(A\cap B)=P(A)\times P(B)$. We have $P(gray)=\frac{10}{20}=0.5$, $P(5) = 0.2$, and $P(gray\cap5)=\frac{2}{20}=0.1$. Also, $P(gray)\times P(5)=0.5\times0.2 = 0.1$. Since $P(gray\cap5)=P(gray)\times P(5)$, the events "gray" and "rank 5" are independent.

Answer:

The probability that a card has rank 5 given it's gray is 0.2. The events "gray" and "rank 5" are independent.