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Answer:
a) W is a binomial random variable because in each trial there are only two possible outcomes, and the probability of a success is always the same.
b) 0.2013 = 20.13% probability that you win exactly 3 times.
Step-by-step explanation:
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
a. Explain why W is a binomial random variable.
For each time a duck is selected, there are only two possible outcomes. Either it has a star, or it does not. Also, after a duck is selected, it is placed back in the water which means that in each trial, the probability of finding a duck with star is the same. This is why W is a binomial random variable.
b. Find the probability that you win exactly 3 times.
Let W=the number of times you win if you play this game 10 times.
This means that [tex]n = 10[/tex]
The game claims to have 20 ducks with stars among the 100 ducks.
This means that [tex]p = \frac{20}{100} = 0.2[/tex]
We have to find P(X = 3). So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 3) = C_{10,3}.(0.2)^{3}.(0.8)^{7} = 0.2013[/tex]
0.2013 = 20.13% probability that you win exactly 3 times.
W is a binomial random variable because in each trial there are only two possible outcomes, and the probability of a success is always the same.
There is 20.13% probability that you win exactly 3 times.
Given that,
In Tsunami Duck Pond there are 100 ducks that get pummeled by tidal waves.
The game claims to have 20 ducks with stars among the 100 ducks.
Let, W = the number of times you win if you play this game 10 times.
We have to determine,
Explain why W is a binomial random variable.
Find the probability that you win exactly 3 times.
According to the question,
- A duck is selected, there are only two possible outcomes. Either it has a star, or it does not. Also, after a duck is selected, it is placed back in the water which means that in each trial, the probability of finding a duck with star is the same. This is why W is a binomial random variable.
- To find the probability that you win exactly 3 times.
Let, W=the number of times you win if you play this game 10 times.
Therefore,
The game claims to have 20 ducks with stars among the 100 ducks.
Then, The probability that you win exactly 3 times is,
[tex]P(X =x) = \ ^nC_x \times P^{x} \times (1-p)^{n-x}[/tex]
When, x = 3 and n = 10
Therefore,
[tex]P(X =3) = \ ^{10}C_3 \times (0.2)^{3} \times (1-0.2)^{10-3}\\\\P(X =3) = \ ^{10}C_3 \times (0.2)^{3} \times (0.8)^{7}\\\\\\P(X =3) = \dfrac{10!}{7!. 3!} \times 0.008 \times 0.20\\\\P(X=3) = 0.2013 = 20.13 \ percent[/tex]
There is 20.13% probability that you win exactly 3 times.
To know more about Probability click the link given below.
https://brainly.com/question/23270039
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