From a single deck of cards, you select a card, look at it, put it back in the deck and shuffle. If you do this 10 times, what is the probability of drawing exactly...
a) 1 Ace
b) 2 Aces
To determine the probability of drawing a certain number of aces from a deck of cards, we need to calculate the number of favorable outcomes (drawing the desired number of aces) divided by the number of possible outcomes (drawing a card each time).
First, let's find out how many aces are in a standard deck of cards. A deck has four aces, one for each suit (hearts, diamonds, clubs, and spades).
a) Probability of drawing exactly 1 Ace:
To calculate this probability, we need to determine the number of ways we can select 1 ace from the deck and multiply it by the number of ways we can select the remaining 9 non-ace cards.
The number of ways to select 1 ace from 4 aces is given by the combination formula: C(4, 1) = 4.
Next, we need to choose 9 non-ace cards from the remaining 48 cards (52 - 4 aces). The number of ways to do this is given by the combination formula: C(48, 9) ≈ 196,947,598.
The total number of possible outcomes is the number of ways to draw 10 cards from a deck of 52: C(52, 10) ≈ 10,861,626,155.
Now, we can calculate the probability of drawing exactly 1 ace:
P(1 Ace) = (number of favorable outcomes) / (number of possible outcomes)
= (C(4, 1) * C(48, 9)) / C(52, 10)
≈ (4 * 196,947,598) / 10,861,626,155
≈ 0.000072 ≈ 0.0072% (rounded to the nearest hundredth)
Therefore, the probability of drawing exactly 1 Ace is approximately 0.0072%.
b) Probability of drawing exactly 2 Aces:
To calculate this probability, we need to determine the number of ways we can select 2 aces from the deck and multiply it by the number of ways we can select the remaining 8 non-ace cards.
The number of ways to select 2 aces from 4 aces is given by the combination formula: C(4, 2) = 6.
Next, we need to choose 8 non-ace cards from the remaining 48 cards. The number of ways to do this is given by the combination formula: C(48, 8) ≈ 79,652,415.
Using the same total number of possible outcomes as before: C(52, 10) ≈ 10,861,626,155.
Now, we can calculate the probability of drawing exactly 2 aces:
P(2 Aces) = (number of favorable outcomes) / (number of possible outcomes)
= (C(4, 2) * C(48, 8)) / C(52, 10)
≈ (6 * 79,652,415) / 10,861,626,155
≈ 0.000343 ≈ 0.0343% (rounded to the nearest hundredth)
Therefore, the probability of drawing exactly 2 Aces is approximately 0.0343%.