Question -
Answer -
(i)
Given a coin is tossedtwice.
Hence, the samplespace of the experiment is S = {HH, HT, TH, TT}
X represents thenumber of heads.
⇒ X (HH) = 2
X (HT) = 1
X (TH) = 1
X (TT) = 0
Therefore, X is afunction on sample space whose range is {0, 1, 2}.
Thus, X is a randomvariable which can take the values 0, 1 or 2.
As we know,
P (HH) = P (HT) = P(TH) = P (TT) = 1/4
P (X = 0) = P (TT) =1/4
P (X = 1) = P (HT) + P(TH) = 1/4 + 1/4 = 1/2
P (X = 2) = P (HH) =1/4
Hence, the requiredprobability distribution is,
| X | 0 | 1 | 2 |
| P (X) | 1/4 | 1/2 | 1/4 |
(ii)
Given three coins aretossed simultaneously. Hence, the sample space of the experiment is S = {HHH,HHT, HTH, THH, TTH, THT, HTT, TTT}
X represents thenumber of tails.
As we see, X is afunction on sample space whose range is {0, 1, 2, 3}.
Thus, X is a randomvariable which can take the values 0, 1, 2 or 3.
P (X = 0) = P(HHH) = 1/8
P (X = 1) = P (HHT) +P (HTH) + P (THH) = 1/8 + 1/8 + 1/8 = 3/8
P (X = 2) = P (HTT) +P (THT) + P (TTH) = 1/8 + 1/8 + 1/8 = 3/8
P (X = 3) = P(TTT) = 1/8
Hence, the required probabilitydistribution is,
| X | 0 | 1 | 2 | 3 |
| P (X) | 1/8 | 3/8 | 3/8 | 1/8 |
(iii)
Given four tosses of acoin.
Hence, the samplespace of the experiment is
S = {HHHH, HHHT, HHTH,HTHH, HTTH, HTHT, HHTT, HTTT, THHH, TTHH, THTH, THHT, THTT, TTHT, TTTH, TTTT}
X represents thenumber of heads.
As we see, X is afunction on sample space whose range is {0, 1, 2, 3, 4}.
Thus, X is a randomvariable which can take the values 0, 1, 2, 3 or 4.
P (X = 0) = P(TTTT) = 1/16
P (X = 1) = P (HTTT) +P (TTTH) + P (THTT) + P (TTHT) = 1/16 + 1/16 + 1/16 + 1/16 = ¼
P(X = 2) = P (HHTT) +P (THHT) + P (TTHH) + P (THTH) + P (HTHT) + P(HTTH)= 1 /16 + 1/16 + 1/16 +1/16 + 1/16 + 1/16 = 6/16 = 3/8
P(X = 3) = P (THHH) +P (HHHT) + P (HTHH) + P (HHTH) = 1/16 + 1/16 + 1/16 + 1/16 = ¼
P(X = 4) = P(HHHH) = 1/16
Hence, the requiredprobability distribution is,
| X | 0 | 1 | 2 | 3 | 4 |
| P (X) | 1/16 | 1/4 | 3/8 | 1/4 | 1/16 |