Respuesta :
Answer:
Step-by-step explanation:
This is a binomial distribution because the probabilities are either that of success or failure.
If probability of success, p = 3/8 = 0.375, then probability of failure, q = 1 - p = 1 - 0.375 = 0.625
The formula is expressed as
P(x = r) = nCr × p^r × q^(n - r)
Where
x represent the number of successes.
p represents the probability of success.
q = represents the probability of failure.
n represents the number of trials or sample.
n = 8
13) P(x = 3) = 8C3 × 0.375^3 × 0.625^(8 - 3) = 0.28
14) P(x = 6) = 8C6 × 0.375^6 × 0.625^(8 - 6) = 0.03
15) P(x = 1) = 8C1 × 0.375^1 × 0.625^(8 - 1) = 0.11
16) P(x = 5) = 8C5 × 0.375^5 × 0.625^(8 - 5) = 0.1
17) P(x ≥ 1) = 1 - P(x < 1)
P(x < 1) = P(x = 0)
P(x = 0) = 8C0 × 0.375^0 × 0.625^(8 - 0) = 0.023
P(x ≥ 1) = 1 - 0.023 = 0.977
18) P(x ≥2) = 1 - P(x < 2)
P(x < 2) = P(x = 0) + P(x = 1)
P(x = 0) = 8C0 × 0.375^0 × 0.625^(8 - 0) = 0.023
P(x = 1) = 8C1 × 0.375^1 × 0.625^(8 - 1) = 0.11
P(x ≥ 2) = 1 - (0.023 + 0.11) = 0.867
19) P(x ≥ 6) = P(x = 6) + P(x = 7) + P(x = 8)
P(x = 6) = 8C6 × 0.375^6 × 0.625^(8 - 6) = 0.03
P(x = 7) = 8C7 × 0.375^7 × 0.625^(8 - 7) = 0.005
P(x = 8) = 8C8 × 0.375^8 × 0.625^(8 - 8) = 0.0004
P(x ≥ 6) = 0.03 + 0.005 + 0.0004 = 0.0354
20) P(x ≥ 7) = P(x = 7) + P(x = 8)
P(x ≥ 7) = 0.005 + 0.0004 = 0.0054
21) P(x ≤ 6) = 1 - P(x = 8)
P(x ≤ 6) = 1 - 0.0004 = 0.9996
22) P(x ≤ 6) = 1 - [P(x = 7) + P(x = 8)]
P(x ≤ 6) = 1 - (0.005 + 0.0004) = 0.9946
The probabilities are listed below:
- The probability of success of exactly 3 successes is 5.27 %.
- The probability of success of exactly 6 successes is 0.28 %.
- The probablity of success of exactly 1 success is 37.5 %.
- The probablity of success of exactly 5 successes is 0.74 %.
- The probability of success of at least 1 success is at most 37.5 %.
- The probability of success of at least 2 successes is at most 14.1 %.
- The probablity of success of at least 6 successes is at most 0.28 %.
- The probability of success of at least 7 successes is at most 0.10 %.
- The probability of success of at most 7 successes is at least 0.10 %.
- The probability of success of at most 6 successes is at least 0.28 %.
If each trial represents an independent event, then the probability of success of a given number of consecutive trials ([tex]p_{T}[/tex]) in defined by the following formula:
[tex]p_{T} = p^{n}[/tex] (1)
Where:
- [tex]p[/tex] - Success probability for a sole event.
- [tex]n[/tex] - Number of consecutive events.
If we know that [tex]p = \frac{3}{8}[/tex], then the probabilities associated to a given number of trials:
1) 3 successes
[tex]p_{3} = \left(\frac{3}{8} \right)^{3}[/tex]
[tex]p_{3} = \frac{27}{512}[/tex]
The probability of success of exactly 3 successes is 5.27 %.
2) 6 successes
[tex]p_{6} = \left(\frac{3}{8} \right)^{6}[/tex]
[tex]p_{6} = \frac{729}{262144}[/tex]
The probability of success of exactly 6 successes is 0.28 %.
3) 1 success
[tex]p_{1} = \frac{3}{8}[/tex]
The probablity of success of exactly 1 success is 37.5 %.
4) 5 successes
[tex]p_{5} = \left(\frac{3}{8} \right)^{5}[/tex]
[tex]p_{5} = \frac{243}{32768}[/tex]
The probablity of success of exactly 5 successes is 0.74 %.
5) At least 1 success
The probability of success of at least 1 success is at most 37.5 %.
6) At least 2 successes
The probability of success of at least 2 successes is at most 14.1 %.
7) At least 6 successes
The probablity of success of at least 6 successes is at most 0.28 %.
8) At least 7 successes
The probability of success of at least 7 successes is at most 0.10 %.
9) At most 7 successes
The probability of success of at most 7 successes is at least 0.10 %.
10) At most 6 successes
The probability of success of at most 6 successes is at least 0.28 %.
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