g) What is the probability of picking a three given that the card is a spade? Round answer to 3 decimal places. $$ \square $$

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UpStudy AI · Accepted Answer

1. Define the conditional probability:  
   The probability of picking a three given that the card is a spade is  
   $$
   P(\text{Three} \mid \text{Spade}) = \frac{P(\text{Three and Spade})}{P(\text{Spade})}.
   $$

2. Calculate the numerator $$ P(\text{Three and Spade}) $$:  
   There is only one three of spades in the deck. Hence,  
   $$
   P(\text{Three and Spade}) = \frac{1}{52}.
   $$

3. Calculate the denominator $$ P(\text{Spade}) $$:  
   There are 13 spades in the deck. Therefore,  
   $$
   P(\text{Spade}) = \frac{13}{52}.
   $$

4. Compute the conditional probability:  
   $$
   P(\text{Three} \mid \text{Spade}) = \frac{\frac{1}{52}}{\frac{13}{52}} = \frac{1}{13}.
   $$

5. Convert the fraction to a decimal and round to 3 decimal places:  
   $$
   \frac{1}{13} \approx 0.0769 \approx 0.077.
   $$

Thus, the probability is $$0.077$$.
The probability of picking a three given that the card is a spade is 0.077.

g) What is the probability of picking a three given that the card is a spade? Round answer to 3 decimal places. \( \square \)

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