Question

Consider a coin whose probability of landing heads is p. For what values of p can you guarantee that the probability of obtaining: (i) at least one heads in 6 tosses is strictly less than .15? (ii) exactly 3 heads in 6 tosses is strictly less than .15?

Answer 1

i) P( at least one heads in 6 tosses)=1-P(no head in 6 tosses)=1-(1-p)⁶ <0.15

(1-p)⁶ >0.85

1-p >0.85^1/6

p <1-0.9733

p < 0.0267

ii)

P(exactly 3 heads in 6 tosses)=⁶C₃p³(1-p)³ <0.15

20*(p(1-p))³ <0.15

p-p² <(0.15/20)^1/3

p²-p+0.195743 >0

solving above :

p <0.2671 or p>0.7329