Theory of Expectation :: Problems on Tossing Coins : Probability Distribution

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Three coins, whose faces are marked 1 and 2, are tossed what is the expectation of total value of numbers on their faces?

Net Answers :
[Expectation: 3 ; Variance:0.5]

Solution

Let "x" indicate the total value of the numbers appearing on the face of the coins
[Since you are required to find the expectation of the total value of numbers, the variable would represent the total of the numbers appearing on the face of the coins.]
The total value of the numbers appearing on the face of the coins would be

4 if both the coins show up 2

3 if one coin shows up 2 and the other 1

2 if both the coins show up 1

⇒ The values carried by the variable ("x") would be either 1, 3, or 4
⇒ "X" is a discrete random variable with range = {1, 3, 4}
"X" represents the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x"
In the experiment of tossing the 2 coins

Total no. of possible choices = 2 × 2

= 4

Let "O" represent the coin showing up 1 and "T" represent the coin showing up 2.
Probabilty that the 2 coins would show up

Two TWO's =
No. of favorable choices
Total no. of possible choices

⇒ P(TT) =
1
4

For the two letter word "TT"
No. of letters
in total = 2 ⇒ n = 2
of the first kind = 2 {T, T} ⇒ a = 2
all of which are different = 0 ⇒ x = 0

No. of favorable choices = No. of ways in which the two
letters of the word TT can be permuted.

=
n!
a!

=
2!
2!

= 1

A "ONE" and a "TWO" =
No. of favorable choices
Total no. of possible choices

⇒ P(OT) =
2
4

For the two letter word "OT"
No. of letters
in total = 2 ⇒ n = 2
all of which are different = 2 {O, T} ⇒ x = 2

No. of favorable choices = No. of ways in which the twoe
letters of the word OT can be permuted.

= n!

= 2!

= 2 × 1

= 2

Two ONE's =
No. of favorable choices
Total no. of possible choices

⇒ P(OO) =
1
4

For the two letter word "OO"
No. of letters
in total = 2 ⇒ n = 2
of the first kind = 2 {O, O} ⇒ a = 2
all of which are different = 0 {} ⇒ x = 0

No. of favorable choices = No. of ways in which the two
letters of the word OO can be permuted.

=
n!
a!

=
2!
2!

= 1

Therefore, probability that the total value of the numbers on the faces of the two coins is

4 ⇒ P(X = 4) =

1
4

3 ⇒ P(X = 3) =

2
4
(Or)
1
2

2 ⇒ P(X = 2) =
1
4

The probabilty distribution of "x" would be

x 2 3 4

P(X = x)
1
4

2
4

1
4

Calculations for Mean and Standard Deviations

x P (X = x) px
[x × P (X = x)] x² px²
[x² × P (X = x)]

2
1
4

2
4
4
4
4

3
2
4

6
4
9
18
4

4
1
4

4
4
16
16
4

Total 1
12
4

38
4

= 3 = 9.5

Expected total value of the numbers on the faces

⇒ Expectation of "x"

⇒ E (x) = Σ px

= 3

Variance of the total value of the numbers on the faces

⇒ var (x) = E (x²) − (E(x))²

⇒ var (x) = Σ px² − (Σ px)²

= 9.5 − (3)²

= 9.5 − 9

= 0.5

Standard Deviation of the total value of the numbers on the faces

⇒ SD (x) = + √ Var (x)

= + √ 0.5

= + 0.707

Credit : Vijayalakshmi Desu