Respuesta :
Answer:
The probability that the coins are thrown more than three times to show the same face is 0.3164.
Step-by-step explanation:
The problem is related to Geometric distribution.
The Geometric distribution defines the probability distribution of X failures before the first success.
The probability distribution function is:
[tex]P(X=k)=(1-p)^{k}p;\ k = 0, 1, 2, ...[/tex]
First compute the probability that in the [tex]i^{th}[/tex] throw all the three coins will show the same face.
P (All the 3 coins shows the same face) = P (All the three coins shows Heads) + P (All the three coins shows Tails)
[tex]=(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} )+(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} )\\=\frac{1}{8}+\frac{1}{8}\\ =\frac{2}{8} \\=\frac{1}{4}[/tex]
Now compute the probability that it takes more than 3 throws for the coins to show the same face.
P (X > 3) = 1 - P (X ≤ 3)
[tex]=1-[P(X=1)+P(X=2)+P(X=3)]\\=1-[[(1-\frac{1}{4} )^{0}\times\frac{1}{4}]+[(1-\frac{1}{4} )^{1}\times\frac{1}{4}]+ [(1-\frac{1}{4} )^{2}\times\frac{1}{4}]+[(1-\frac{1}{4} )^{3}\times\frac{1}{4}]]\\=1-[0.2500+0.1875+0.1406+0.1055]\\=1-0.6836\\=0.3164[/tex]
Thus, the probability that it takes more than 3 throws for the coins to show the same face is 0.3164.
The probability that it takes more than 3 throws for the coins to show the same face is 0.3164.
What is probability?
Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
Given
Three identical fair coins are thrown simultaneously until all three show the same face.
Then the total event will be
[tex]\rm Total \ event = 2^3 = 8[/tex]
All the possibilities.
[tex]\rm (HHH),( HHT ),(HTH),( THH),(HTT),(THT),(TTH),(TTT)[/tex]
For getting the same face, then the probability will be
[tex]\rm P(same\ face) = \dfrac{2}{8} = \dfrac{1}{4}[/tex]
Now compute the probability that it takes more than 3 throws for the coins to show the same face.
[tex]\rm P (X > 3) = 1 - P (X ≤ 3)\\\\P (X > 3) = 1 -[P(x=1) +P(x=2) + P(x=3)]\\\\P (X > 3) =1 -\{[(1-\dfrac{1}{4})^0 * \dfrac{1}{4}] +[(1-\dfrac{1}{4})^1 * \dfrac{1}{4}] +[(1-\dfrac{1}{4})^2 * \dfrac{1}{4}] +[(1-\dfrac{1}{4})^3 * \dfrac{1}{4}] \}\\\\P (X > 3) =1 - 0.6836\\\\P (X > 3) =0.3164[/tex]
Thus, the probability that it takes more than 3 throws for the coins to show the same face is 0.3164.
More about the probability link is given below.
https://brainly.com/question/795909
