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Five mathematical expectations
The basic method of calculating expectation is to multiply each possible value by the probability of taking that value, and then add all the products. Details are as follows:

The first question:

Expected return =-2× 0.99+(50-2 )× 0.01=-1.5.

The second question:

The expected income of single lottery =-2× 0.9329+(14.9-2 )× 0.0671=-1.00021.

Expected value of 10 Note =10×-1.0001=-10.5438+0.

The third question:

The expected income of single-note lottery = -5×0.98+(50-5)×0.02 = -4.

2 note that expected value =2× -4 = -8.

The fourth question:

* * * There are 3×3=9 throwing methods.

The number of letters in mailbox A is 0, and there are 2×2=4 ways to send letters.

The letter number of mailbox A is 1, and there are 2×2=4 voting methods.

When the number of letters is 2, there are 1 ways to throw out the letters in mailbox A.

eζ= 0×4/9+ 1×4/9+2× 1/9 = 6/9

The fifth question:

The probability of zeta = 2 is 0.9.

The probability of zeta =1is 0. 1×0.9=0.09.

The probability of zeta = 0 is1-0.1-0.09 = 0.01.

eζ= 2×0.9+ 1×0.09+0×0.0 1 = 1.89