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For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal places) is _____
Correct answer is between '15,16'. Can you explain this answer?
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For a given biased coin, the probability that the outcome of a toss is...
Given Information:
- Probability of getting a head in a single toss of the biased coin = 0.4
- The coin is tossed 1,000 times
Random Variable and its Definition:
- Let X be the random variable representing the number of times a head appears in the 1,000 tosses.
Mean of X:
- The mean of a binomial distribution is given by the formula: Mean (μ) = n * p
- Here, n = 1,000 (number of trials) and p = 0.4 (probability of success)
- Therefore, the mean of X is μ = 1,000 * 0.4 = 400
Variance of X:
- The variance of a binomial distribution is given by the formula: Variance (σ^2) = n * p * (1 - p)
- Here, n = 1,000 (number of trials) and p = 0.4 (probability of success)
- Therefore, the variance of X is σ^2 = 1,000 * 0.4 * (1 - 0.4) = 240
Standard Deviation of X:
- The standard deviation (σ) of a random variable is the square root of its variance.
- Therefore, the standard deviation of X is σ = √(240) ≈ 15.49
Answer and Explanation:
- The standard deviation of X is approximately 15.49, which rounds to 15.50 when rounded to 2 decimal places.
- As the correct answer is between 15 and 16, the answer is indeed correct.
- The standard deviation measures the spread or dispersion of the random variable X.
- In this case, it tells us how much the actual number of heads in the 1,000 tosses is likely to deviate from the mean of 400.
- A standard deviation of 15.50 indicates that the number of heads is likely to be within 1 standard deviation (15.50) of the mean (400) about 68% of the time.
- Therefore, we can expect the number of heads to fall within the range of 384.50 to 415.50 (400 ± 15.50) about 68% of the time.
- This provides us with a measure of the variability or uncertainty in the outcomes of the 1,000 tosses.
- Probability of getting a head in a single toss of the biased coin = 0.4
- The coin is tossed 1,000 times
Random Variable and its Definition:
- Let X be the random variable representing the number of times a head appears in the 1,000 tosses.
Mean of X:
- The mean of a binomial distribution is given by the formula: Mean (μ) = n * p
- Here, n = 1,000 (number of trials) and p = 0.4 (probability of success)
- Therefore, the mean of X is μ = 1,000 * 0.4 = 400
Variance of X:
- The variance of a binomial distribution is given by the formula: Variance (σ^2) = n * p * (1 - p)
- Here, n = 1,000 (number of trials) and p = 0.4 (probability of success)
- Therefore, the variance of X is σ^2 = 1,000 * 0.4 * (1 - 0.4) = 240
Standard Deviation of X:
- The standard deviation (σ) of a random variable is the square root of its variance.
- Therefore, the standard deviation of X is σ = √(240) ≈ 15.49
Answer and Explanation:
- The standard deviation of X is approximately 15.49, which rounds to 15.50 when rounded to 2 decimal places.
- As the correct answer is between 15 and 16, the answer is indeed correct.
- The standard deviation measures the spread or dispersion of the random variable X.
- In this case, it tells us how much the actual number of heads in the 1,000 tosses is likely to deviate from the mean of 400.
- A standard deviation of 15.50 indicates that the number of heads is likely to be within 1 standard deviation (15.50) of the mean (400) about 68% of the time.
- Therefore, we can expect the number of heads to fall within the range of 384.50 to 415.50 (400 ± 15.50) about 68% of the time.
- This provides us with a measure of the variability or uncertainty in the outcomes of the 1,000 tosses.
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Community Answer
For a given biased coin, the probability that the outcome of a toss is...
Data
p = probability of success event = 0.4
q = probability of failure event = 0.6
n = Number of trials
Formulae:
Standard Deviation = 
Variance in Binomial Distribution: npq
Calculation:
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For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal places) is _____Correct answer is between '15,16'. Can you explain this answer?
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