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Math  /  Data & Statistics

QuestionA certain businessman visits various companies, looking for investors for his startup. It is known that the chance that a potential investor will not decide to engage is 35.5%35.5 \%. We assume that the decisions of potential investors are independent. The businessman continues his visits until the third refusal (i.e. until he sees the third person who decides not to invest). Let XX denote the number of companies visited by the investor. Calculate P(X=10)P(X=10).
Round the result to THREE decinnal places.

Studdy Solution

STEP 1

1. The probability of a potential investor not engaging is p=0.355 p = 0.355 .
2. The decisions of potential investors are independent.
3. The businessman stops visiting after the third refusal.
4. X X is the number of companies visited until the third refusal.

STEP 2

1. Identify the probability distribution.
2. Set up the probability formula for the distribution.
3. Calculate P(X=10) P(X=10) .

STEP 3

The problem describes a negative binomial distribution because we are counting the number of trials until a fixed number of failures (3 refusals) occurs.

STEP 4

The probability mass function for a negative binomial distribution is given by:
P(X=k)=(k−1r−1)(1−p)rpk−r P(X = k) = \binom{k-1}{r-1} (1-p)^r p^{k-r}
where r r is the number of failures (3 refusals), p p is the probability of success (0.645), and k k is the total number of trials (10 visits).

STEP 5

Substitute the values into the formula:
P(X=10)=(10−13−1)(0.355)3(0.645)10−3 P(X = 10) = \binom{10-1}{3-1} (0.355)^3 (0.645)^{10-3}
P(X=10)=(92)(0.355)3(0.645)7 P(X = 10) = \binom{9}{2} (0.355)^3 (0.645)^7
Calculate the binomial coefficient:
(92)=9×82×1=36 \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
Calculate P(X=10) P(X = 10) :
P(X=10)=36×(0.355)3×(0.645)7 P(X = 10) = 36 \times (0.355)^3 \times (0.645)^7
P(X=10)=36×0.0447×0.0651 P(X = 10) = 36 \times 0.0447 \times 0.0651
P(X=10)=36×0.00291197 P(X = 10) = 36 \times 0.00291197
P(X=10)=0.104791 P(X = 10) = 0.104791
Round the result to three decimal places:
P(X=10)≈0.105 P(X = 10) \approx 0.105
The probability P(X=10) P(X = 10) is:
0.105 \boxed{0.105}

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