Let probability of getting head =x Then, probability of getting tail =1−x Tom wins if he gets head H, TTH, TTTTH, ... (alternative through) Probability of Tom winning game =62.5%=
625
1000
P(H)+P(TTH)+P(TTTTH)+...=
625
1000
x+(1−x)(1−x)x+(1−x)4x+...=
625
1000
⇒x[1+(1−x)2+(1−x)4+...]=
625
1000
⇒x+
1
1−(1−x)2
=
625
100
⇒
x
2x−x2
=
625
1000
⇒
1
2−x
=
625
1000
⇒x=
2
5
( probability of head ) ⇒1−x=
3
5
(probability of tails) ∴ Probability of getting exactly 3 heads on tossing a coin or 5 times =5C3x3(1−x)2=10×(