NCERT Solutions
Class 12 Maths
Probability
Ex.13.4 Q.3
Find the probability distribution of
(i) number of heads in two tosses of a coin
(ii) number of tails in the simultaneous tosses of three coins
(iii) number of heads in four tosses of a coin
(i) When one coin is tossed twice, the sample space is {HH, HT, TH, TT}
Let X represent the number of heads.
So, X(HH) = 2, X(HT) = 1, X(TH) = 1, X(TT) = 0
Therefore, X can take the value of 0, 1, or 2.
It is known that,
P(HH) = 2, P(HT) = 1, P(TH) = 1, P(TT) = 
P (X = 0) = P (TT) =
P (X = 1) = P (HT) + P (TH) 
P (X = 2) = P (HH) = 
Thus, the required probability distribution is as follows:
|
X |
0 |
1 |
2 |
|
P(X) |
¼ |
1/2 |
1/4 |
(ii) When three coins are tossed simultaneously, the sample space is
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Let X represent the number of tails.
It can be seen that X can take the value of 0, 1, 2, or 3.
P(X = 0) = P(HHH) = 
P(X = 1) = P(HHT) + P(HTH) + P(THH) 
P(X = 2) = P(HTT) + P(THT) + P(TTH)
P(X = 3) = P(TTT) = 18
Thus, the probability distribution is as follows:
|
X |
0 |
1 |
2 |
3 |
|
P(X) |
1/8 |
3/8 |
3/8 |
1/8 |
(iii) When a coin is tossed four times, the sample space is
S = HHHH, HHHT, HHTH, HHTT, HTHT, HTHH, HTTH, HTTT,
THHH, THHT, THTH, THTT, TTHH, TTTH, TTTH, TTTT
Let X be the random variable, which represents the number of heads.
It can be seen that X can take the value of 0, 1, 2, 3, or 4.
P(X = 0) = P(TTTT) = 1/16
P(X = 1) = P(TTTH) + P(TTHT) + P(THTT) + P(HTTT)
P(X = 2) = P(HHTT) + P(THHT) + P(TTHH) + P(HTTH) + P(HTHT) + P(THTH)
P (X = 3) = P (HHHT) + P (HHTH) + P (HTHH) P (THHH)
P (X = 4) = P (HHHH) 
Thus, the probability distribution is as follows:
|
X |
0 |
1 |
2 |
3 |
4 |
|
P(x) |
1/16 |
1/4 |
3/8 |
1/4 |
1/16 |