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.
Answers
2. 6 times
3. 12 times
Solution:-
i) When a coin is tossed twice, then the sample space is
➠S=(HH,HT,TH,TT), which contains four equally likely sample points.
Let X denote the number of heads in any outcome S ,
Then
➝ X(HH)=2,X(HT)=1,X(TH)=1 and X(TT)=0
➝ P(X=0) =P (tails occurs, on both tosses)= P=(TT)=1/4
➝ P(X=1)=P (one head and one tail occurs )=P(TH,HT)=2/4=1/2
➝ P(X=2)=P ( head occurs on both tosses ) = P(HH)=1/4
Probablity of distribution X is
X ➝ 0 1 2
P(X)➝ 1/4 1/2 1/4
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
ii) When a coin is tossed thrice ,the sample space is
➠S=(HHH,HHT,HTH,HTT.THH,THT,TTH,TTT)
which contains eight equally likely sample points .
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
Let X denote the number of tails in any outcome ω∈S ,then X can take value 0,1,2 and 3.
➝P(X=0)=P( no tail ) =P(HHH) =1/8
➝P(X=1)=P (one tails two heads show up) =P(HHT,HTH,THH)=3/8
➝P(X=2)=P( two tail amd one head show up ) =P(HTT,THT,TTH)=3/6
➝P(X=3)=P( three tails show up) =P(TTT)=1/8
Probablity of distribution X is
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, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT)
which contains 16 equally likely sample points
Let X denote the number of heads in any outcome ω∈S ,then X can take values of 0,1,2,3 and 4.
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
➝P(X=0)=P ( no head shows up ) =P(TTTT)=1/16
➝P(X=1)=P (one head and three tails shows up ) =P(HTTT,THTT,TTHT,TTTH)=4/16=1/4
➝P(X=2)=P( two head and two tails shows up )=P(HHTT,TTHH,HTHT,HTTH,THHT,THTH,)=6/16=3/8
➝P(X=3)=P( three heads and one tails shows up )= P (HHHT,HHTH,HTHH,THHH)=4/16=1/4:
➝P(X=4)=P (four heads show up ) =P(HHHH)=1/16
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
Probablity of distribution X is
X ➝ 0 1 2 3 4
P(X)➝ 1/16 1/4 3/16 1/4 1/16