Question

Probability that missing card is not a spade, given two drawn cards are spades?

• A. $\frac{3}{50}$
• B. $\frac{39}{50}$
• C. $\frac{39}{52}$
• D. $\frac{38}{52}$

Hint:

Bayes’ Theorem. Update prior probabilities using evidence from observed events.

Answer 1

The Correct Option is B

Use Bayes’ theorem with two cases: missing card is spade or not. \[ P(E|M_S) = \frac{66}{1275},\quad P(E|M_{NS}) = \frac{78}{1275} \] \[ P(M_{NS}|E) = \frac{78 \cdot 39}{78 \cdot 39 + 66 \cdot 13} = \frac{39}{50} \]

Answer 2

Step 1: Understand the problem
A standard deck has 52 cards with 13 spades. Two cards are drawn and both are spades. We need to find the probability that a missing (third) card is not a spade.

Step 2: Cards left after drawing two spades
After removing two spades, the deck has \(52 - 2 = 50\) cards left.
Remaining spades = \(13 - 2 = 11\)
Remaining non-spades = \(50 - 11 = 39\)

Step 3: Calculate the probability
The missing card is one of these 50 cards. Probability that this card is not a spade:
\[ P(\text{not a spade}) = \frac{\text{number of non-spade cards left}}{\text{total cards left}} = \frac{39}{50} \]

Final answer: \(\displaystyle \frac{39}{50}\)