Answer:
The solution is attached in the pictures below
Step-by-step explanation:
Answer:
Probability that [tex]X_{90}[/tex] is at least 160 is 0.0127
Step-by-step explanation:
Probability that [tex]X_{90}[/tex] is at least 160
[tex]P(X_{90} \geq 160) = 1 - P(X_{90}\leq160)[/tex]
[tex]P(X_{90}\leq160) = P(\frac{X_{90}- \mu }{\sigma}\leq \frac{160- \mu }{\sigma})[/tex]
[tex]Probability = \frac{number of possible outcomes}{number of total outcomes}[/tex]
Probability that she rolls 1 or 2 i.e. probability that she takes one step to the right:
P(X=1) = 2/6 = 1/3
Probability that she rolls 3,4,5,6 i.e. Probability that she takes two steps to the right:
P(X=2) = 4/6 = 2/3
[tex]\mu = E(X) = \sum xP(X)\\[/tex]
when x = 1,2
[tex]\mu = E(X) = (1*\frac{1}{3} )+(2*\frac{2}{3} )[/tex]
[tex]\mu = \frac{5}{3}[/tex]
The mean value after n flips
[tex]\mu_{90} = \frac{5}{3} * 90\\\mu_{90} = 150[/tex]
For the standard deviation:
[tex]\sigma_{90} =\sqrt{ [E(x^{2}) -((E(x))^{2} ]*90} \\\sigma_{90} =\sqrt{ [(1^{2}*\frac{1}{3})+(2^{2}*\frac{2}{3} ) -(\frac{5}{3}) ^{2} ]*90}[/tex]
[tex]\sigma_{90} = \sqrt{(3-\frac{25}{3})*90 } \\\sigma_{90} = 4.47[/tex]
[tex]P(X_{90}\leq160) = P(Z\leq \frac{160- 150 }{4.47})[/tex]
Where Z = [tex]\frac{X_{90}- \mu }{\sigma}[/tex]
[tex]P(X_{90}\leq160) = P(Z\leq 2.24)[/tex] = 0.9873
[tex]P(X_{90} \geq 160) = 1 - 0.9873\\P(X_{90} \geq 160) =0.0127[/tex]
