FIND THE PROBABILITY OF DRAWING AT LEAST ONE CLUB WHEN YOU DRAW CARDS FROM A STANDAD 52-CARD DECK 52 TIMES; ASSUME YOU REPLACE THE CARD EAC TIMEYOU DRAW, SO THERE ARE ALWAYS 52 CARDS O DRAW FROM. LIST THE APPROPRIE FORMULA, TEN COMPUTE YOUR ANSWER.
13/52
Well, it's time for a little math humor! Probability, eh? Let me entertain you with a humorous response before we get to the serious stuff.
Why did the playing card go to therapy?
Because it was feeling a bit "shuffled" about its identity!
Now, let's dive into the problem. We want to find the probability of drawing at least one club when the cards are replaced each time. The probability of drawing a club in a single draw is 13/52 (since there are 13 clubs in a deck of 52 cards).
To find the probability of drawing at least one club in 52 draws, we can use the complementary probability. The complementary probability is equal to 1 minus the probability of the event not happening. So, the formula is:
P(at least one club) = 1 - P(no clubs in 52 draws)
The probability of not drawing a club in a single draw is 39/52 (since there are 39 non-club cards left after drawing a club). Using the concept of independent events, the probability of not drawing a club in 52 draws is (39/52) raised to the power of 52:
P(no clubs in 52 draws) = (39/52)^52
Now, let's put this all together and calculate:
P(at least one club) = 1 - (39/52)^52
P(at least one club) ≈ 0.997
So, there's a roughly 99.7% chance of drawing at least one club in 52 draws from a shuffled 52-card deck, with replacement after each draw.
To find the probability of drawing at least one club when you draw cards from a standard 52-card deck 52 times, you can use the concept of complementary events.
The probability of drawing at least one club can be found by subtracting the probability of not drawing any clubs from 1.
The formula to calculate the probability of drawing at least one club is:
P(at least one club) = 1 - P(no clubs)
Now, let's calculate the probability.
Step 1: Calculate the probability of not drawing any clubs in a single draw.
Since there are 13 clubs in a deck of 52 cards, the probability of not getting a club on a single draw is:
P(no clubs) = (52 - 13)/52 = 39/52 = 3/4
Step 2: Calculate the probability of not drawing any clubs in 52 draws.
Since each draw is independent and the cards are replaced after each draw, the probability of not drawing any clubs in all 52 draws is the same as the probability of not drawing any clubs in a single draw, raised to the power of the number of draws:
P(no clubs in 52 draws) = (3/4)^52
Step 3: Calculate the probability of drawing at least one club in 52 draws.
Using the formula mentioned earlier:
P(at least one club) = 1 - P(no clubs in 52 draws) = 1 - (3/4)^52
Now, let's compute the answer:
P(at least one club) = 1 - (3/4)^52
Using a calculator or a mathematical software, the approximate value of P(at least one club) is approximately 0.99999999999999997864, which is essentially equivalent to 1.
Therefore, the probability of drawing at least one club when you draw cards from a standard 52-card deck 52 times is nearly 1 or 100%.
The probability ofdrawing just one club in the 52 draws = 1/4 * (3/4)^51
The probability of drawing two clubs — which fits the "at least" description — in 52 draws = (1/4)^2 * (3/4)^50
The probability of drawing three clubs in 52 draws = (1/4)^3 * (3/4)^49
Or you could go from the opposite direction, 1 - probability of drawing no clubs.
Does that lead you to a formula?
I hope this helps. Thanks for asking.