Answer:
Explanation:
Binomial experiments are modeled by the formula:
[tex]P(X=x)=C(n,x)\cdot p^x\cdot (1-p)^{(n-x)}[/tex]
Where
- P(X=x) is the probability of exactly x successes
- [tex]C(n,x)=\dfrac{n!}{x!\cdot (n-x)!}[/tex]
- p is the probability of one success, which must be the same for every trial, and every trial must be independent of other trial.
- n is the number of trials
- 1 - p is the probability of fail
- there are only two possible outcomes for each trial: success or fail.
a.) P (x=1)
[tex]P(X=1)=\dfrac{6!}{1!\cdot (6-1)!}\times (0.3)^1\times(1-0.3)^{(6-1)}=0.302526[/tex]
b.) P (x=5)
[tex]P(X=5)=\dfrac{5!}{5!\cdot (6-5)!}\times (0.3)^5\times (1-0.3)^{(6-5)}=0.010206[/tex]
c.) P (x=3)
Using the same formula:
[tex]P(X=3)=0.18522[/tex]
d.) P (x less than or equal to 3)
- P(X≤3)= P(X=3) + P(X=2) + P(X=1) + P(X=0)
Also,
- P(X≤3) = 1 - P(X≥4) = 1 - P(X=4) - P(X=5) - P(X=6)
You can use either of those approaches. The result is the same.
Using the second one:
- P(X=4) = 0.059335
- P(X=5) = 0.010206
- P(X=6) = 0.000729
- P(X≤3) = 1 - 0.05935 - 0.010206 - 0.000729 = 0.92953
e.) P(x greather than or equal to 5)
- P(X≥5) = 0.010206 + 0.000729 = 0.010935
f.) P(x less than or equal 4)
- P(X≤4) = 1 - P(X≥5) = 1 - P(X=5) - P(X=6)
- P(X≤4) = 1 - 0.010206 - 0.000729 = 0.989065
