5 coins are tossed 3200times. Find the expected frequency for heads and tails also find the mean and varience of the number of heads?
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To find the expected frequency for heads and tails, we need to know the probability of getting a head or a tail when flipping a fair coin. Since there are two possible outcomes (head or tail) and each outcome has an equal chance of occurring, the probability of getting a head or a tail is 0.5 or 1/2.
Now, let's calculate the expected frequency for heads and tails based on the number of times the coins are tossed.
Expected Frequency for Heads:
Expected Frequency = Total Number of Tosses x Probability of Getting a Head
= 3200 × 0.5
= 1600
Expected Frequency for Tails:
Expected Frequency = Total Number of Tosses x Probability of Getting a Tail
= 3200 × 0.5
= 1600
Therefore, the expected frequency for heads is 1600 and for tails is also 1600.
Next, let's calculate the mean and variance of the number of heads.
Mean:
The mean is the average value of a set of data, and it is calculated by multiplying the probability of an event by the number of times it occurs, and then summing up these products. In this case, the probability of getting a head is 0.5, and the number of times it occurs is the number of tosses (3200).
Mean = Probability of Getting a Head × Total Number of Tosses
= 0.5 × 3200
= 1600
Therefore, the mean number of heads is 1600.
Variance:
The variance measures the spread or dispersion of a set of data. In this case, we need to calculate the variance of the number of heads. The formula for variance is the sum of the squared differences between each value and the mean, divided by the total number of values.
Variance = (Σ(x - μ)²) / n
where Σ denotes the sum, x denotes the number of heads, μ denotes the mean, and n denotes the total number of tosses.
To calculate the variance, we need to find the squared difference between each number of heads and the mean, and then sum up these squared differences.
Let's assume the number of heads to be x1, x2, x3, x4, x5, ..., xn (where n = 3200).
Variance = [(x1 - μ)² + (x2 - μ)² + (x3 - μ)² + (x4 - μ)² + ... + (xn - μ)²] / n
Since all the coins are tossed independently, the number of heads in each toss follows a binomial distribution with p = 0.5.
The variance of a binomial distribution is given by the formula:
Variance = n × p × (1 - p)
Therefore, the variance of the number of heads is:
Variance = 3200 × 0.5 × (1 - 0.5)
= 1600
Therefore, the variance of the number of heads is 1600.