Question

1. Three fair dices were rolled.
(a) How many possible outcomes there will be, if the number in each dice was recorded and the order of dices are considered.
(b) How many possible outcomes there will be, if the sum of the dices are recorded.
(c) What is the probability of getting a result with the sum of the three dices exactly equals to 6?
(d) What is the probability of getting a result with the sum of the three dices less than 6?
2. Seven fair coins were flipped and there outcomes of each coin (head or tail) were recorded.
(a) How many possible outcomes there will be, if the order of coins are considered?
(b) How many possible outcomes with exactly 3 heads(and 4 tails)?
(c) What is the probability of getting a result with exactly 3 heads (and 4 tails)?
(d) What is the probability of getting a result with less than 2 head(and more than 5 tails)?

Answer 1

1.

(a)

Since there are 6 outcomes(1,2,3,4,5,6) for each dice and we have three dice then,

Possible outcomes there will be, if the number in each dice was recorded = 6*6*6 = 216

(b)

Suppose we get 1 on each dice

Sum of 1 on each dice = 3

Possible combinations : (1,1,1) = 1

Similarly suppose we get a combination like : '1' appearing on two dice and '2' appearing on one dice.

Sum = 4

Possible combinations : (1,1,2) , (1,2,1), (2,1,1) = 3

and So on.......

We get the following table :

S.No.	TOTAL	COMBINATIONS	PROBABILITY
1	3	1	0.0046
2	4	3	0.0139
3	5	6	0.0278
4	6	10	0.0463
5	7	15	0.0694
6	8	21	0.0972
7	9	25	0.1157
8	10	27	0.1250
9	11	27	0.1250
10	12	25	0.1157
11	13	21	0.0972
12	14	15	0.0694
13	15	10	0.0463
14	16	6	0.0278
15	17	3	0.0139
16	18	1	0.0046
	Total	216	1

Possible outcomes there will be, if the sum of the dices are recorded = 16

(c)

We can refer the table in the previous part :

Probability of getting a result with the sum of the three dices exactly equals to 6

= Number of combinations when the sum of the three dices exactly equals to 6 / Total combinations

= 21/216 = 0.0972

(d)

Probability of getting a result with the sum of the three dices less than 6

=  Probability of getting a result with the sum of the three dices as 3,4 and 5

= Probability of getting a result with the sum of the three dices as 3 + Probability of getting a result with the sum of the three dices as 4 + Probability of getting a result with the sum of the three dices as 5

Probability of getting a result with the sum of the three dices as 3

= Number of combinations when sum of the three dices is 3 / Total combinations

= 1 / 216 = 0.0046

Probability of getting a result with the sum of the three dices as 4

= Number of combinations when sum of the three dices is 4 / Total combinations

= 3 / 216 = 0.0139

Probability of getting a result with the sum of the three dices as 5

= Number of combinations when sum of the three dices is 5 / Total combinations

= 6 / 216 = 0.0278

Probability of getting a result with the sum of the three dices less than 6 = 0.0046 + 0.0139 + 0.0278 = 0.0463

2.

(a)

Since there are 2 outcomes (Head(H) or Tail(T)) for each coin and 7 coins are flipped then,

Possible outcomes there will be, if the order of coins are considered = 2⁷ = 128

(b)

Suppose there are no heads in 7 coins flipped

So, Combinations : (T,T,T,T,T,T,T)= 1

Suppose there is 1 head in 7 coins flipped

So, Combinations : (H,T,T,T,T,T,T), (T,H,T,T,T,T,T), (T,T,H,T,T,T,T),(T,T,T,H,T,T,T),(T,T,T,T,H,T,T),(T,T,T,T,T,H,T),(T,T,T,T,T,T,H)  = 7

And So on.......

Let h = number of heads

Formula becomes : 7!/ ( h! *(7-h)! )

n! = n*(n-1)*(n-2)*............*1

Number of Heads	Combinations
0	1
1	7
2	21
3	35
4	35
5	21
6	7
7	1
Total	128

Possible outcomes with exactly 3 heads = 35

(c)

Probability of getting a result with exactly 3 heads = Possible outcomes with exactly 3 heads / Total outcomes =

= 35 / 128 = 0.273

(d)

Probability of getting a result with less than 2 head = Probability of getting a result with 0 heads +  Probability of getting a result with 1 head

Probability of getting a result with 0 heads =  Possible outcomes with 0 heads / Total outcomes = 1/128 = 0.0078

Probability of getting a result with 1 head = Possible outcomes with 1 head / Total outcomes = 7/128 = 0.0547

Probability of getting a result with less than 2 heads = 0.0078 + 0.0547 = 0.0625