Question

A.) State the random Variable: select from one of the following - X= the number of heads observed - X=tossing a coin - X= number of coins tossed - X= the number of heads observed when you flip a coin three times - X= the probability that you observe heads B.) construct a probability distribution table for the number of heads obtained over three tosses. Enter the X values from smallest to largest C.) determine the shape of the probability distribution of x - left skewed -symmetric -right skewed - uniform D.) find the MEAN number of heads for this distribution E.) find the standard deviation for the number of heads for this distribution F.) find the probability of obtaining two or less heads over three tosses of a coin

Answer 1

Step 1:

In this question, we are given the following:

Step 2:

A.) State the random variable:

X= the number of heads observed when you flip a coin three times

Step 3:

B.) construct a probability distribution table for the number of heads obtained over three tosses. Enter the X values from smallest to largest:

X P(X)

0 1/8

1 3/8

2 3/8

3 1/8

Step 4:

C.) Determine the shape of the probability distribution of x

- symmetric

Step 5:

D.) find the MEAN number of heads for this distribution

`\begin{gathered} Mean\text{= ( 0 X }(1)/(8))\text{ + ( 1 x }(3)/(8))\text{ + ( 2 x }(3)/(8))\text{ + (3 x }(1)/(8)) \\ \text{Mean = 0+ }(3)/(8)+(6)/(8)+(3)/(8) \\ \text{Mean = }(12)/(8) \\ \text{Mean = 1. 5} \end{gathered}`

Step 6:

E.) find the standard deviation for the number of heads for this distribution:

`S\tan dard\text{ Deviation = }\sqrt[]{(x\text{ -}\mu)^2\text{ P ( X = x )}}`
`\begin{gathered} \sin ce\text{ }\mu\text{ = 1. 5, then we have that:} \\ \sqrt[]{\lbrack(0-1.5)^2X\text{ }(1)/(8)\rbrack+\lbrack(1\text{ - }1.5)^2\text{X }(3)/(8)\rbrack+\lbrack(2-1.5)^2}X(3)/(8)\rbrack+\lbrack(3-1.5)^2\text{ X }(1)/(8) \end{gathered}`
`\begin{gathered} \sqrt[]{(2.\text{ 25 X }(1)/(8))\text{ + ( 0.25 X }(3)/(8))\text{ + ( 0. 25 X }(3)/(8))\text{ + (2.25 X }(1)/(8))} \\ =\text{ }\sqrt[]{0.28125+\text{ 0.09375 + 0.09375 + 0.28125}} \\ =\sqrt[]{0.75} \\ =0.866\text{ ( 3 decimal places)} \end{gathered}`



Step 7:

F.) find the probability of obtaining two or less heads over three tosses of a coin

`P\text{ ( obtaining two or less heads) = }(3)/(8)+\text{ }(3)/(8)+(1)/(8)\text{ = }(7)/(8)`