All the jacks, queens and kings are removed from a deck of 52 playing cards and then well shuffled. Then one card is drawn at random. If an ace is given a value I, find the probability that the card has a value:
(i) 5. (ii) less than 5. (iii) greater than 5.

Answer

To solve the problem step by step, let's analyze the situation and calculate the probabilities for each case.

### Given:
- A standard deck of 52 playing cards has been modified by removing all jacks, queens, and kings.
- This means we remove 12 cards (4 jacks, 4 queens, and 4 kings).
- Therefore, the remaining number of cards = 52 - 12 = 40 cards.
- The values of the cards are from Ace (1) to 10.

### We need to find the probability for three cases:

#### (i) Probability that the card has a value of 5.

1. **Identify the favorable outcomes**:
- There is 1 card of value 5 in each suit (Diamonds, Hearts, Spades, Clubs).
- Therefore, the total number of cards with a value of 5 = 4.

2. **Total outcomes**:
- The total number of remaining cards = 40.

3. **Calculate the probability**:
\[ P(\text{value is 5}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{40} = \frac{1}{10} \]

#### (ii) Probability that the card has a value less than 5.

1. **Identify the favorable outcomes**:
- The values less than 5 are Ace (1), 2, 3, and 4.
- For each of these values, there are 4 cards (one from each suit).
- Therefore, the total number of cards with values less than 5 = 4 (Ace) + 4 (2) + 4 (3) + 4 (4) = 16.

2. **Total outcomes**:
- The total number of remaining cards = 40.

3. **Calculate the probability**:
\[ P(\text{value is less than 5}) = \frac{16}{40} = \frac{2}{5} \]

#### (iii) Probability that the card has a value greater than 5.

1. **Identify the favorable outcomes**:
- The values greater than 5 are 6, 7, 8, 9, and 10.
- For each of these values, there are 4 cards (one from each suit).
- Therefore, the total number of cards with values greater than 5 = 4 (6) + 4 (7) + 4 (8) + 4 (9) + 4 (10) = 20.

2. **Total outcomes**:
- The total number of remaining cards = 40.

3. **Calculate the probability**:
\[ P(\text{value is greater than 5}) = \frac{20}{40} = \frac{1}{2} \]

### Summary of Probabilities:
- (i) Probability of drawing a card with value 5: \( \frac{1}{10} \)
- (ii) Probability of drawing a card with value less than 5: \( \frac{2}{5} \)
- (iii) Probability of drawing a card with value greater than 5: \( \frac{1}{2} \)