Would you be more likely to get at least 70% tails if you flip a fair coin 10 times or if you flip a fair
coin 1000 times?
A) You would be more likely to get at least 70% tails if you flip a fair coin 10 times than if you
flip a fair coin 1000 times.
B) The probability of getting at least 70% tails would be the same regardless of whether the coin
is flipped 10 or 1000 times.
C) You would be more likely to get at least 70% tails if you flip a fair coin 1000 times than if you
flip a fair coin 10 times.
D) There is not enough information to determine which would be more likely.
To determine which option is correct, we need to calculate the probabilities of getting at least 70% tails in each scenario.
Option A states that you would be more likely to get at least 70% tails if you flip a fair coin 10 times than if you flip a fair coin 1000 times.
To calculate the probability of getting at least 70% tails in 10 coin flips, we can use the binomial probability formula. The formula is: P(X ≥ k) = 1 - ΣP(X = i) for i = 0 to k-1. In this case, X represents the number of tails, and k represents the minimum number of tails required to meet the condition.
Let's calculate:
P(X ≥ 7) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6))
Using the binomial probability formula, where n is the number of trials (coin flips), p is the probability of success (50% for a fair coin), and q is the probability of failure (50% for a fair coin), the individual probabilities can be calculated:
P(X = 0) = (10 choose 0) * (0.5)^(0) * (0.5)^(10-0)
P(X = 1) = (10 choose 1) * (0.5)^(1) * (0.5)^(10-1)
P(X = 2) = (10 choose 2) * (0.5)^(2) * (0.5)^(10-2)
...
P(X = 6) = (10 choose 6) * (0.5)^(6) * (0.5)^(10-6)
Now, let's calculate the probability:
P(X ≥ 7) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6))
Similarly, we can calculate the probability of getting at least 70% tails in 1000 coin flips using the same formula with adjusted values:
P(X ≥ 700) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 699))
Now, we compare the calculated probabilities.
If the probability of getting at least 70% tails in the scenario with 10 coin flips is higher than the probability in the scenario with 1000 coin flips, Option A is correct.
If the probabilities are the same, Option B is correct, and if we cannot determine which is more likely, Option D is correct.
By performing the calculations, we can determine the correct option.