# calculating the expected value of random variable, which is net income

there is 1000 lots in the lottery. you can win 1 unit of 100£, 10 units of 50£ and 15 units of 20£. One lot costs 1£. calculate the expected value of random variable, which is net income.

here is my answer: the correct answer is suppose to be -0.1£. what am I missing thanks and sorry for the messiness

• You have calculated the expectation under the condition that you win. This condition must be left out. Unfortunately it is far more likely that you loose. – drhab Oct 11 '14 at 12:49
• ok so how should I go about doing so? – user2864059 Oct 11 '14 at 12:51
• $1$ must not be subtracted from $20$ but from the whole sum of three terms that come in front of $-1$. Is everything clear now? – drhab Oct 11 '14 at 13:09

## 2 Answers

Calculate:

$$\frac{1}{1000}\times100+\frac{10}{1000}\times50+\frac{15}{1000}\times20-1$$

The sum of positive terms is the expectation of the winning and $1$ must be subtracted as price of the lot. You could also write:$$\frac{1}{1000}\times\left(100-1\right)+\frac{10}{1000}\times\left(50-1\right)+\frac{15}{1000}\times\left(20-1\right)+\frac{974}{1000}\times\left(-1\right)$$

• You are very welcome. – drhab Oct 11 '14 at 13:09

Another way to see this is: If you bought all the lots, you would pay $1000$ units and win $1\times 100+10\times 50+15\times 20=900$ units. Thus you would have a net loss of $100$ units for $1000$ lots. So your average (expected value) would be $-\frac{100}{1000}=-0.1$ units per lot.