A bag contains a large number of coins. 60% are 50p coins. 20% are 20p coins. 20% are 10p coins.

Find the mean and the variance of the values of this population of coins.

mean = .6 * 50 + .2 * 20 + .2 * 10 = 30 + 4 + 4 = 38 p

variance = (30-38)^2 + (4-38)^2 + (4-38)^2

To find the mean and variance of the values of this population of coins, we need to calculate the mean and variance separately for each type of coin (50p, 20p, and 10p) and then sum them up.

Let's start with the mean:

For 50p coins:
The proportion of 50p coins is 60%, which means that 60% of the total number of coins are 50p coins.
So, the mean for 50p coins is (0.60 * 50p) = 30p.

For 20p coins:
The proportion of 20p coins is 20%, which means that 20% of the total number of coins are 20p coins.
So, the mean for 20p coins is (0.20 * 20p) = 4p.

For 10p coins:
The proportion of 10p coins is 20%, which means that 20% of the total number of coins are 10p coins.
So, the mean for 10p coins is (0.20 * 10p) = 2p.

Now, let's calculate the overall mean by summing up the mean of each coin type:
Mean = (mean of 50p coins) + (mean of 20p coins) + (mean of 10p coins)
Mean = 30p + 4p + 2p
Mean = 36p

Now, let's move on to the variance:

For 50p coins:
The variance for 50p coins is (0.60 * (50p - 36p)^2).

For 20p coins:
The variance for 20p coins is (0.20 * (20p - 36p)^2).

For 10p coins:
The variance for 10p coins is (0.20 * (10p - 36p)^2).

Now, let's calculate the overall variance by summing up the variances of each coin type:
Variance = (variance for 50p coins) + (variance for 20p coins) + (variance for 10p coins)

Note: To calculate the variance, we need to convert pence to pounds by dividing by 100, as variance is usually expressed in square units.

Once you have calculated the individual variances for each coin type using the formula described above, you can sum them up to obtain the overall variance of the population.

To find the mean and variance of the values of this population of coins, we need to calculate the weighted average of the coin values, where the weights are the percentages.

Let's assume that there are 100 coins in the bag to keep calculations simple.

Number of 50p coins = (60/100) * 100 = 60 coins
Number of 20p coins = (20/100) * 100 = 20 coins
Number of 10p coins = (20/100) * 100 = 20 coins

Now, let's calculate the total value of the coins and the mean:

Total value of 50p coins = 60 coins * 50p = £30
Total value of 20p coins = 20 coins * 20p = £4
Total value of 10p coins = 20 coins * 10p = £2

Total value of all the coins = £30 + £4 + £2 = £36

Mean = Total value of all the coins / Total number of coins
Mean = £36 / 100 = £0.36

So, the mean value of the population of coins is £0.36.

Now, let's calculate the variance.

Variance is a measure of how spread out the values are from the mean. To calculate the variance, we need to calculate the squared difference between each coin value and the mean, and then take the weighted average of those squared differences.

Squared difference from mean for 50p coins = (0.50 - 0.36)^2 * 60 = 0.0216 * 60 = 1.296
Squared difference from mean for 20p coins = (0.20 - 0.36)^2 * 20 = 0.0256 * 20 = 0.512
Squared difference from mean for 10p coins = (0.10 - 0.36)^2 * 20 = 0.1296 * 20 = 2.592

Total variance = Sum of squared differences / Total number of coins
Total variance = (1.296 + 0.512 + 2.592) / 100 = 4.4 / 100 = 0.044

So, the variance of the values of this population of coins is 0.044.