Answer :
Answer:
a.[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}\\[/tex]
b. [tex]E(x) = 0.3[/tex]
c. [tex]S(x)=0.5196[/tex]
d. [tex]E=5,000[/tex]
Step-by-step explanation:
The probability that the company won x bids follows a binomial distribution because we have n identical and independent experiments with a probability p of success and (1-p) of fail.
So, the PMF of X is equal to:
[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}\\[/tex]
Where p is 0.1 and it is the chance of winning. Additionally, n is 3 and it is the number of bids. So the PMF of X is:
[tex]P(x)=\frac{3!}{x!(3-x)!}*0.1^{x}*(1-0.1)^{n-x}\\[/tex]
For binomial distribution:
[tex]E(x)=np\\S(x)=\sqrt{np(1-p)}[/tex]
Therefore, the company can expect to win 0.3 bids and it is calculated as:
[tex]E(x) = np = 3*0.1 = 0.3[/tex]
Additionally, the standard deviation of the number of bids won is:
[tex]S(x)=\sqrt{np(1-p)}=\sqrt{3(0.1)(1-0.1)}=0.5196[/tex]
Finally, the probability to won 1, 2 or 3 bids is equal to:
[tex]P(1)=\frac{3!}{1!(3-1)!}*0.1^{1}*(1-0.1)^{3-1}=0.243\\P(2)=\frac{3!}{2!(3-2)!}*0.1^{2}*(1-0.1)^{3-2}=0.027\\P(3)=\frac{3!}{3!(3-3)!}*0.1^{3}*(1-0.1)^{3-3}=0.001[/tex]
So, the expected profit for the company is equal to:
[tex]E=-10,000+50,000(0.243)+100,000(0.027)+150,000(0.001)\\E=5,000[/tex]
Because there is a probability of 0.243 to win one bid and it will produce 50,000 of income, there is a probability of 0.027 to win 2 bids and it will produce 100,000 of income and there is a probability of 0.001 to win 3 bids and it will produce 150,000 of income.