Answer:
[tex]P(X<0.1)= 5.5x10^{-5}[/tex]
Step-by-step explanation:
Previous concepts
Beta distribution is defined as "a continuous density function defined on the interval [0, 1] and present two parameters positive, denoted by α and β, both parameters control the shape. "
The probability function for the beta distribution is given by:
[tex] P(X)= \frac{x^{\alpha-1} (1-x)^{\beta -1}}{B(\alpha,\beta)}[/tex]
Where B represent the beta function defined as:
[tex]B(\alpha,\beta)= \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}[/tex]
Solution to the problem
For our case our random variable is given by:
[tex] X \sim \beta (\alpha=5, \beta =2)[/tex]
We can use the following R code to plot the distribution for this case:
> x=seq(0,1,0.01)
> plot(x,dbeta(x,5,2),main = "Beta distribution a=5, b=2",ylab = "Probability")
And we got as the result the figure attached.
And for this case we want this probability, since we want the probability that she has at most 10% or 0.1 change of winning:
[tex]P(X<0.1)[/tex]
And we can find this probability with the following R code:
> pbeta(0.1,5,2)
[1] 5.5e-05
And we got then this : [tex]P(X<0.1)= 5.5x10^{-5}[/tex]
