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Two unbiased coins are tossed together five times. What is the probability that heads turn up on both the coins for three times?
- a)9/512
- b)27/512
- c)45/512
- d)63/512
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Two unbiased coins are tossed together five times. What is the probab...
The probability that heads turn up when two coins are tossed = (1/2)(1/2) = ¼
∴ The probability that three times heads turn up on both the coins
= 5C3 (1/4)3 (1-1/4)2
= (10) (1/64)(9/16) = 45/512
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Two unbiased coins are tossed together five times. What is the probab...
Understanding the Problem
Two unbiased coins are tossed together five times. We need to find the probability that heads turn up on both coins exactly three times.
Defining the Outcomes
- When tossing two coins, the possible outcomes are:
- HH (both heads)
- HT (heads on first, tails on second)
- TH (tails on first, heads on second)
- TT (both tails)
- The probability of getting heads on both coins (HH) in one toss is 1/4.
Using Binomial Probability Formula
- We can model this situation using the binomial probability formula:
P(X = k) = (n choose k) * (p^k) * (q^(n-k))
Where:
- n = total number of trials (5)
- k = number of successful trials (3)
- p = probability of success (1/4)
- q = probability of failure (3/4)
Calculating the Values
- n choose k (5 choose 3) = 5! / (3! * 2!) = 10
- p^k = (1/4)^3 = 1/64
- q^(n-k) = (3/4)^(5-3) = (3/4)^2 = 9/16
Putting It All Together
- Now substitute the values into the formula:
P(X = 3) = 10 * (1/64) * (9/16)
- Calculate:
- 10 * 1/64 = 10/64 = 5/32
- (5/32) * (9/16) = (45/512)
Final Result
Therefore, the probability that heads turn up on both coins exactly three times is 45/512, which is option 'C'.
Two unbiased coins are tossed together five times. We need to find the probability that heads turn up on both coins exactly three times.
Defining the Outcomes
- When tossing two coins, the possible outcomes are:
- HH (both heads)
- HT (heads on first, tails on second)
- TH (tails on first, heads on second)
- TT (both tails)
- The probability of getting heads on both coins (HH) in one toss is 1/4.
Using Binomial Probability Formula
- We can model this situation using the binomial probability formula:
P(X = k) = (n choose k) * (p^k) * (q^(n-k))
Where:
- n = total number of trials (5)
- k = number of successful trials (3)
- p = probability of success (1/4)
- q = probability of failure (3/4)
Calculating the Values
- n choose k (5 choose 3) = 5! / (3! * 2!) = 10
- p^k = (1/4)^3 = 1/64
- q^(n-k) = (3/4)^(5-3) = (3/4)^2 = 9/16
Putting It All Together
- Now substitute the values into the formula:
P(X = 3) = 10 * (1/64) * (9/16)
- Calculate:
- 10 * 1/64 = 10/64 = 5/32
- (5/32) * (9/16) = (45/512)
Final Result
Therefore, the probability that heads turn up on both coins exactly three times is 45/512, which is option 'C'.
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Two unbiased coins are tossed together five times. What is the probability that heads turn up on both the coins for three times?a)9/512b)27/512c)45/512d)63/512Correct answer is option 'C'. Can you explain this answer?
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