Question

if we draw a card from those above, whats the probability that it has rank 5? (enter as a decimal.) which of the options represents the likelihood of drawing a card that is rank 5, given that its gray. p(gray | rank 5) 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.) are events \gray\ and
ank 5\ independent? no, since knowing the co

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).

Step3: Calculate probability of drawing card with rank 5

Probability $P(\text{rank }5)=\frac{\text{Number of cards with rank 5}}{\text{Total number of cards}}=\frac{4}{20} = 0.2$.

Step4: Count number of gray - colored cards

There are 10 gray - colored cards.

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

There are 2 gray - colored cards with rank 5 (Z5, W5).

Step6: Calculate conditional probability

The conditional probability $P(\text{rank }5|\text{gray})=\frac{\text{Number of gray cards with rank 5}}{\text{Number of gray cards}}=\frac{2}{10}=0.2$.

Step7: Check for independence

Two events $A$ and $B$ are independent if $P(A|B)=P(A)$. Here, $P(\text{rank }5) = 0.2$ and $P(\text{rank }5|\text{gray})=0.2$. But we should also check $P(\text{gray}|\text{rank }5)$. There are 4 cards with rank 5 and 2 of them are gray, so $P(\text{gray}|\text{rank }5)=\frac{2}{4} = 0.5$ and $P(\text{gray})=\frac{10}{20}=0.5$. Since $P(\text{rank }5|\text{gray}) = P(\text{rank }5)$ and $P(\text{gray}|\text{rank }5)=P(\text{gray})$, the events "gray" and "rank 5" are independent.

Answer:

The probability that a drawn card has rank 5 is 0.2. The correct option for the likelihood of drawing a card that is rank 5 given that it's gray is $P(\text{rank }5|\text{gray})$. The probability that a drawn gray - colored card has rank 5 is 0.2. The events "gray" and "rank 5" are independent.