Albert Stanley
4 days ago
What is the probability that at least two heads are obtained when a coin is tossed 3 times?
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ChatGPT
4 days ago
To find the probability of getting at least two heads when a coin is tossed 3 times, we need to calculate the probability of getting exactly 2 heads and the probability of getting exactly 3 heads, and then add them together.
The probability of getting exactly 2 heads in 3 coin tosses can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (3 coin tosses)
- k is the number of successful outcomes (2 heads)
- p is the probability of success on a single trial (0.5 for a fair coin)
P(X=2) = (3 choose 2) * (0.5)^2 * (0.5)^(3-2)
P(X=2) = 3 * 0.25 * 0.5
P(X=2) = 0.375
The probability of getting exactly 3 heads in 3 coin tosses is:
P(X=3) = (3 choose 3) * (0.5)^3 * (1-0.5)^(3-3)
P(X=3) = 1 * 0.125 * 1
P(X=3) = 0.125
Therefore, the probability of getting at least two heads when a coin is tossed 3 times is:
P(at least 2 heads) = P(X=2) + P(X=3)
P(at least 2 heads) = 0.375 + 0.125
P(at least 2 heads) = 0.5
So, the probability of getting at least two heads when a coin is tossed 3 times is 0.5 or 50%.
The probability of getting exactly 2 heads in 3 coin tosses can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (3 coin tosses)
- k is the number of successful outcomes (2 heads)
- p is the probability of success on a single trial (0.5 for a fair coin)
P(X=2) = (3 choose 2) * (0.5)^2 * (0.5)^(3-2)
P(X=2) = 3 * 0.25 * 0.5
P(X=2) = 0.375
The probability of getting exactly 3 heads in 3 coin tosses is:
P(X=3) = (3 choose 3) * (0.5)^3 * (1-0.5)^(3-3)
P(X=3) = 1 * 0.125 * 1
P(X=3) = 0.125
Therefore, the probability of getting at least two heads when a coin is tossed 3 times is:
P(at least 2 heads) = P(X=2) + P(X=3)
P(at least 2 heads) = 0.375 + 0.125
P(at least 2 heads) = 0.5
So, the probability of getting at least two heads when a coin is tossed 3 times is 0.5 or 50%.
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Google Gemini AI
4 days ago
The probability of getting at least two heads when a coin is tossed 3 times is 0.75.
Here is the breakdown of all possible outcomes:
HHH (all heads): 1/8
HHT (2 heads, 1 tail): 3/8
HTH (2 heads, 1 tail): 3/8
THH (2 heads, 1 tail): 3/8
HHTT (1 head, 2 tails): 3/8
HTTH (1 head, 2 tails): 3/8
THHT (1 head, 2 tails): 3/8
TTT (all tails): 1/8
As you can see from the breakdown, there are 7 out of 8 possible outcomes that result in at least two heads. Therefore, the probability of getting at least two heads is 7/8 = 0.75.
Here is the breakdown of all possible outcomes:
HHH (all heads): 1/8
HHT (2 heads, 1 tail): 3/8
HTH (2 heads, 1 tail): 3/8
THH (2 heads, 1 tail): 3/8
HHTT (1 head, 2 tails): 3/8
HTTH (1 head, 2 tails): 3/8
THHT (1 head, 2 tails): 3/8
TTT (all tails): 1/8
As you can see from the breakdown, there are 7 out of 8 possible outcomes that result in at least two heads. Therefore, the probability of getting at least two heads is 7/8 = 0.75.
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