Respuesta :
Answer:
The probability of getting 4 or more days when the surf is at least 6 feet is 0.544
Step-by-step explanation:
We can modelate this exercise as a binomial random variable.
The probability function of a binomial random variable is :
[tex]P(X=r)=(nCr)p^{r}(1-p)^{n-r}[/tex]
Where [tex]P(X=r)[/tex] is the probability of the variable X to assume the value r
nCr is the combinatorial number define as [tex]nCr=\frac{n!}{r!(n-r)!}[/tex]
n is the number of binomial experiments that we make. In our exercise, n is the number of random days we pick of January.
And finally p is the success probability.
In our exercise, we define X : ''The number of days when the surf is at least 6 feet''.
And we are looking for P(X≥4).
P(X≥4) = P(X = 4) + P(X = 5) + P(X = 6)
[tex]P(X=4)=6C4(0.6^{4})(0.4^{2})[/tex]
[tex]P(X=5)=6C5(0.6^{5})(0.4^{1})[/tex]
[tex]P(X=6)=6C6(0.6^{6})(0.4^{0})=0.6^{6}[/tex]
Finally
P(X≥4)=[tex]6C4(0.6^{4})(0.4^{2})+6C5(0.6^{5})(0.4^{1})+0.6^{6}=0.544[/tex]
P(X≥4) = 0.544
Probability of some event represents the change of occurrence of that event.
The probability of getting 4 or more days when the surf is at least 6 feet is 0.544 approx.
How to calculate the probability of an event?
Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.
Then, suppose we want to find the probability of an event E.
Then, its probability is given as
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]
Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.
How to find that a given condition can be modeled by binomial distribution?
Binomial distributions consists of n independent Bernoulli trials.
Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))
Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as
[tex]X \sim B(n,p)[/tex]
The probability that out of n trials, there'd be x successes is given by
[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]
For the given case, let we have:
E = event of a surf being of at least 6 feet.
Then we have:
[tex]P(E) = 60\% = 0.6[/tex]
and
[tex]P(E') = P(\text{surf less than 6 feet}) = 1 - 0.6 = 0.4[/tex] (since its complement of E)
Now, since we have surf on all 6 days independent of each other, and they are either at least of 6 feet(success)(probability is 0.6) or less than 6 feet (failure) (probability is 0.4)
Thus, this condition can be modeled by binomial distribution, where
- X = number of surfs of at least 6 feet
- n = 6
- p = 0.6
- 1- p = q = 0.4
Thus, [tex]X \sim B(6,0.6)[/tex]
The needed probability is [tex]\begin{aligned} P(X \geq 4) &= P(X = 4) + P(X = 5) + P(X = 6) \\\\&= \:^6C_4(0.6)^4(0.4)2 + \:^6C_5(0.6)^5(0.4)^1 + \:^6C_6(0.6)^6(0.4)^0 \\&=0.31104 + 0.186624 + 0.046656\\&\approx0.544\\\end{aligned}[/tex]
Thus,
The probability of getting 4 or more days when the surf is at least 6 feet is 0.544 approx.
Learn more about binomial distribution here:
https://brainly.com/question/13609688
