Two pennies one with Phead u and one with Phead w are to b

Two pennies, one with P(head) = u and one with P(head) = w, are to be tossed. The outcomes are independent of each other. Define: P0= P(0 heads occur), P1= P(1 head occurs), P2=P(2 Heads occur). Can u and w be chosen such that P0=P1=P2? Yes or no? Prove your answer

Solution

a) P(0) = w²

b) P(1) = 2uw

c) P(2) = u²

where u+w =1

P0=P1=P2 means

w² = u² =2uw

If we take the first two we get w =u = 1/2

Then 2uw = 1/2

Hence possible

u = w = 1/2 will satisfy this.