The following table contains the probability distribution for X = the number data packages transmitted through satellite

What is the probability (between 0 and 1 with 2 decimal places) of at least one successful transmission?

X

0

1

2

3

p(X)

0.35

0.35

0.25

0.05

Question

The following table contains the probability distribution for X = the number data packages transmitted through satellite

What is the probability (between 0 and 1 with 2 decimal places) of at least one successful transmission?

X

0

1

2

3

p(X)

0.35

0.25

0.05

BlackTom AI · Accepted Answer

"At least one successful transmission" means X ≥ 1, which is the complement of X = 0. Basic probability rules: probabilities for all possible outcomes sum to 1, and P(A or B or C) for mutually exclusive outcomes equals the sum of their probabilities. The complement rule states P(X ≥ 1) = 1 − P(X = 0).

Step-by-step reasoning:
1. List given probabilities: P(0) = 0.35, P(1) = 0.35, P(2) = 0.25, P(3) = 0.05.
2. Use the complement approach: P(at least one) = 1 − P(0).
3. Compute: 1 − 0.35 = 0.65.

Alternate (equivalent) approach by direct summation:
P(X ≥ 1) = P(1) + P(2) + P(3) = 0.35 + 0.25 + 0.05.
Intermediate additions: 0.35 + 0.25 = 0.60; 0.60 + 0.05 = 0.65.
Both methods yield 0.65.

Why this is correct:
- The only way to have zero successful transmissions is the single event X = 0 with probability 0.35. All other outcomes (1, 2, or 3 successes) are mutually exclusive and collectively represent "at least one," so summing their probabilities or using the complement gives the correct result.
- The probability values provided add to 1 (0.35 + 0.35 + 0.25 + 0.05 = 1), confirming a valid distribution.

Why other potential answers would be wrong:
- Using 0.35 would mistakenly give P(0) instead of P(at least one).
- Any number other than 0.65 (for example adding incorrectly or forgetting one term) would violate the complement rule or the requirement that probabilities sum correctly.

Conclusion: The probability of at least one successful transmission is 0.65, which matches the provided answer.

The following table contains the probability distribution for X = the number data packages transmitted through satellite What is the probability (between 0 and 1 with 2 decimal places) of at least one successful transmission? X 0 1 2 3 p(X) 0.35 0.35 0.25 0.05数值题

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