Respuesta :
Answer:
Probability of rolling at least 4 sixes is 0.01696.
Step-by-step explanation:
We are given that an unbalanced die is manufactured so that there is a 20% chance of rolling a “six." The die is rolled 6 times.
The above situation can be represented through binomial distribution;
[tex]P(X = r) = \binom{n}{r} \times p^{r} \times (1-p)^{n-r};x=0,1,2,3,.......[/tex]
where, n = number trials (samples) taken = 6 trials
r = number of success = at least 4
p = probability of success which in our question is probability of
rolling a “six", i.e; p = 0.20
Let X = Number of sixes on a die
So, X ~ Binom(n = 6, p = 0.20)
Now, Probability of rolling at least 4 sixes is given by = P(X [tex]\geq[/tex] 4)
P(X [tex]\geq[/tex] 4) = P(X = 4) + P(X = 5) + P(X = 6)
= [tex]\binom{6}{4} \times 0.20^{4} \times (1-0.20)^{6-4}+\binom{6}{5} \times 0.20^{5} \times (1-0.20)^{6-5}+\binom{6}{6} \times 0.20^{6} \times (1-0.20)^{6-6}[/tex]
= [tex]15 \times 0.20^{4} \times 0.80^{2}+6 \times 0.20^{5} \times 0.80^{1}+1 \times 0.20^{6} \times 0.80^{0}[/tex]
= 0.0154 + 0.00154 + 0.000064
= 0.01696
Therefore, probability of rolling at least 4 sixes is 0.01696.
Using the binomial distribution, it is found that there is a 0.0086 = 0.86% probability of rolling at least 4 sixes.
For each roll, there are only two possible outcomes, either it is a six, or it is not. The result of a roll is independent of any other rolls, hence, the binomial distribution is used to solve this question.
Binomial probability distribution
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem:
- The die is rolled 6 times, hence [tex]n = 6[/tex].
- There are six sides on the dice, one of which is 6, hence [tex]p = \frac{1}{6} = 0.1667[/tex].
The probability is:
[tex]P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{6,4}.(0.1667)^{4}.(0.8333)^{2} = 0.0080[/tex]
[tex]P(X = 5) = C_{6,5}.(0.1667)^{5}.(0.8333)^{1} = 0.0006[/tex]
[tex]P(X = 6) = C_{6,6}.(0.1667)^{6}.(0.8333)^{0} \approx 0[/tex]
Then:
[tex]P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.0080 + 0.0006 + 0 = 0.0086[/tex]
0.0086 = 0.86% probability of rolling at least 4 sixes.
For more on the binomial distribution you can check https://brainly.com/question/24863377
