Let's go through each part of the problem step by step.

Information about the Deck

• A standard deck has 52 cards.
• Cards are ranked from 2 through 10, Jack, Queen, King, and Ace.
• There are 4 suits (hearts, diamonds, clubs, and spades), with each suit having 13 cards.

Part (a): Probability of a card under a 4

Cards under a 4 are: 2, and 3 (Ace is considered high here). Each of these ranks has 4 cards (one per suit), so:

• Total cards under a 4 = $2 \times 4 = 8$
• Probability = $\frac{8}{52} = \frac{2}{13}$

Part (b): Probability of a card above an 8

Cards above an 8 are: 9, 10, Jack, Queen, King, and Ace. Each rank has 4 cards, so:

• Total cards above an 8 = $6 \times 4 = 24$
• Probability = $\frac{24}{52} = \frac{6}{13}$

Part (c): Probability of a card both under a 4 and above an 8

The events "under a 4" and "above an 8" are mutually exclusive (they cannot happen at the same time). Therefore:

• Probability = $0$

Part (d): Probability of a card either under a 4 or above an 8

We use the addition rule for probabilities:

• Probability(under a 4) = $\frac{2}{13}$
• Probability(above an 8) = $\frac{6}{13}$
• Since these two events are mutually exclusive, the probability of either event occurring is: $\frac{2}{13} + \frac{6}{13} = \frac{8}{13}$

Summary of Answers

• (a) $\frac{2}{13}$
• (b) $\frac{6}{13}$
• (c) $0$
• (d) $\frac{8}{13}$

Would you like further details or have any questions?

Related Questions

1. What is the probability of drawing a face card (Jack, Queen, or King) from the deck?
2. What is the probability of drawing a red card (hearts or diamonds)?
3. What is the probability of drawing a card that is either an Ace or a King?
4. How many ways can you draw a card that is a spade or a number less than 5?
5. What is the probability of drawing a card that is either black or an even number?

Tip

When dealing with probabilities involving cards, always verify whether events are mutually exclusive to correctly apply the addition rule.