70 fair coins are tossed independently, and 210 fair dice are rolled independent

Question

70 fair coins are tossed independently, and 210 fair dice are rolled independently. Let C be the number of coins that land showing "heads" on top and let D be the number of dice that land showing an even number of spots on top. The expected value of C, E(C), is (Q8) The expected value of D, E(D), is (Q9) The chance that C is within +2 of E(C) is (Q10) 2 The chance that C/70 is between 40% and 60% is (Q11) the chance that D is within ± 2 of E(D). ? the chance that D210 is between 40% and 60%. Problem 6 Three dice are rolled and two fair coins are tossed. Let X be the sum of the number of spots that show on the top faces of the dice and the number of coins that land heads up. The expected value of X is (Q12)

Explanation / Answer

Problem 6

Solution:
Dice have numbers 1 to 6, the average is (1+2+3+4+5+6) = 21/6 = 3.5.

Thrown 3 times, the average is 3 x 3.5 = 10.5

The chance of heads is 0.5, thrown 2 times the probability is 0.5 x 2 = 1.

So the expectation is 10.5 + 1 = 11.5.

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