Math

Question Select the mutually exclusive option: P(red) and P(black)P(red) \text{ and } P(black) from a standard deck, or P(sum=6) and P(sum=4)P(sum = 6) \text{ and } P(sum = 4) from two dice rolls.

Studdy Solution

STEP 1

Assumptions
1. A standard deck of cards has 52 cards, with 26 red cards and 26 black cards.
2. A standard die has 6 faces with numbers 1 through 6.
3. Mutually exclusive events are events that cannot occur at the same time.
4. "With replacement" means that after drawing a card, it is placed back into the deck before the next draw.
5. "Without replacement" would mean that the card is not placed back into the deck after the draw.

STEP 2

Analyze the first option: drawing a red card and then drawing a black card with replacement from a standard deck of cards.

STEP 3

Since the card is replaced after the first draw, the outcome of the second draw is independent of the first draw. Therefore, these events are not mutually exclusive; it is possible to draw a red card first and then a black card.

STEP 4

Analyze the second option: rolling a sum of 6 from the first two rolls of a standard die and a sum of 4 from the second two rolls.

STEP 5

The sum of 6 from the first two rolls and the sum of 4 from the second two rolls are dependent on the outcome of the second roll. However, it is possible to achieve both events with different outcomes for the second roll (e.g., first roll is a 5, second roll is a 1 for the sum of 6; then the second roll is a 3, and the third roll is a 1 for the sum of 4). Therefore, these events are not mutually exclusive.

STEP 6

Analyze the third option: rolling a sum greater than 10 from two rolls of a standard die and rolling a 3 for the first roll.

STEP 7

To achieve a sum greater than 10, the possible outcomes for two rolls are (5, 6), (6, 5), and (6, 6). If the first roll is a 3, it is impossible to achieve a sum greater than 10 with the second roll. Therefore, these events are mutually exclusive.

STEP 8

Analyze the fourth option: drawing a 7 and then drawing another 7 with replacement from a standard deck of cards.

STEP 9

Since the card is replaced after the first draw, it is possible to draw a 7 and then draw another 7. Therefore, these events are not mutually exclusive.

STEP 10

Based on the analysis, the option that represents mutually exclusive events is the third one: rolling a sum greater than 10 from two rolls of a standard die and rolling a 3 for the first roll.
The correct answer is: rolling a sum greater than 10 from two rolls of a standard die and rolling a 3 for the first roll.

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