Finding Probabilities of Compound Events
A regular deck of cards has 52 cards with 4 aces. Carl asked a friend to pick a card from the deck three times, replacing the card each time. His friend picked three aces. Which expression will give the probability of that event?
$\left(\frac{4}{52}\right)\left(\frac{4}{52}\right)\left(\frac{4}{52}\right)$
$\left(\frac{1}{52}\right)\left(\frac{1}{51}\right)\left(\frac{1}{50}\right)$
$\left(\frac{4}{52}\right)\left(\frac{4}{51}\right)\left(\frac{4}{50}\right)$
$\left(\frac{4}{52}\right)\left(\frac{3}{51}\right)\left(\frac{2}{50}\right)$
Answer (2)
The probability of drawing an ace from a deck of 52 cards is 52 4 .
Since the card is replaced each time, the draws are independent events.
The probability of three independent events occurring in sequence is the product of their probabilities: 52 4 × 52 4 × 52 4 .
The expression for the probability is: ( 52 4 ) ( 52 4 ) ( 52 4 ) .
Explanation
Understand the problem We are given a problem about finding the probability of a compound event. Specifically, we want to find the probability of drawing an ace from a deck of cards three times in a row, with replacement. This means that after each draw, the card is put back into the deck, so the probabilities for each draw remain the same.
Determine the probability of drawing an ace A standard deck of cards has 52 cards, and 4 of them are aces. Therefore, the probability of drawing an ace on any single draw is 52 4 . Since we replace the card each time, the draws are independent events.
Calculate the probability of three consecutive aces To find the probability of three independent events occurring in sequence, we multiply their individual probabilities. In this case, the probability of drawing an ace three times in a row with replacement is: 52 4 × 52 4 × 52 4 .
State the final answer The expression that gives the probability of picking an ace three times in a row with replacement is ( 52 4 ) ( 52 4 ) ( 52 4 ) .
Examples
This type of probability calculation is useful in many real-world scenarios, such as determining the likelihood of winning a lottery or predicting the outcome of a series of independent events. For example, if you were playing a game where you needed to roll a specific number on a die three times in a row, you could use this method to calculate the probability of success. Let's say you want to roll a 6 on a six-sided die three times in a row. The probability of rolling a 6 on one die is 6 1 . Therefore, the probability of rolling a 6 three times in a row is 6 1 × 6 1 × 6 1 = 216 1 .
The probability of drawing three aces from a deck of cards, when each card is replaced, is calculated by multiplying the probabilities of drawing an ace each time. The correct expression for this is ( 52 4 ) ( 52 4 ) ( 52 4 ) . Therefore, the selected option is the first expression provided.
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