if we draw a card from those above, whats the probability that it has rank 5? (enter as a decimal.) which of…

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.