A machine flips 4 coins. The probability that the [math: i]th coin lands heads is [math: pi], where:[math: p1=0.5] [math: p2=0.7] [math: p3=0.8] [math: p4=0.9] What is the expected value of the total number of heads observed?[Fill in the blank]单项选择题
A
a. 2.9
B
b. 0.252
C
c. 2
D
d. 0.725

Question

A machine flips 4 coins. The probability that the [math: i]th coin lands heads is [math: pi], where:[math: p1=0.5] [math: p2=0.7] [math: p3=0.8] [math: p4=0.9] What is the expected value of the total number of heads observed?[Fill in the blank]单项选择题

A

a. 2.9

B

b. 0.252

C

c. 2

D

d. 0.725

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
Random outcomes for each coin can be represented by indicator random variables Xi where Xi = 1 if coin i shows heads and 0 otherwise. The expectation of an indicator is E[Xi] = P(Xi = 1) = pi. Linearity of expectation states that the expectation of a sum of random variables equals the sum of their expectations, regardless of independence.

2. Step-by-step reasoning
Define X = total number of heads = X1 + X2 + X3 + X4. By linearity of expectation,
E[X] = E[X1] + E[X2] + E[X3] + E[X4].

3. Calculations with intermediate steps
E[X1] = p1 = 0.5
E[X2] = p2 = 0.7
E[X3] = p3 = 0.8
E[X4] = p4 = 0.9

Sum these:
E[X] = 0.5 + 0.7 + 0.8 + 0.9
First 0.5 + 0.7 = 1.2
Then 1.2 + 0.8 = 2.0
Finally 2.0 + 0.9 = 2.9

4. Why the correct answer is correct
Because each coin contributes its probability of heads to the expected total and expectations add, the total expected number of heads is simply the sum of the four given probabilities, 2.9. This directly matches option a.

5. Why the other options are incorrect
b (0.252) equals the product 0.5·0.7·0.8·0.9, which is the probability that all four coins are heads, not the expected count. c (2) would be the total if each coin had average probability 0.5, but here probabilities are higher. d (0.725) equals the average probability per coin (2.9/4), not the expected total number of heads.

6. Conclusion
Therefore the expected total number of heads is 2.9, confirming the correct answer is a. 2.9.

A machine flips 4 coins. The probability that the [math: i]th coin lands heads is [math: pi], where:[math: p1=0.5] [math: p2=0.7] [math: p3=0.8] [math: p4=0.9] What is the expected value of the total number of heads observed?[Fill in the blank]单项选择题

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