Question

A coin is tossed twice. Let Z denote the number of heads on

the first toss and W , the total number of heads on the two tosses. If

the coin is unbalanced and a head has a 60% chance of occurring,

find

(a) the joint distribution of W and Z .

(b) the marginal distribution of W .

(c )the marginal distribution of Z .

(d) the probability that at least 1 head occurs. 

Answer 1

$a:\\Z\,\,takes\,\,values\,\,0,1\\W\,\,takes\,\,values\,\,0,1,2\\P\left( Z=0,W=0 \right) =0.4\cdot 0.4=0.16\\P\left( Z=0,W=1 \right) =0.4\cdot 0.6=0.24\\P\left( Z=1,W=1 \right) =0.6\cdot 0.4=0.24\\P\left( Z=1,W=2 \right) =0.6\cdot 0.6=0.36\\b:\\P\left( W=0 \right) =0.4\cdot 0.4=0.16\\P\left( W=1 \right) =0.4\cdot 0.6+0.6\cdot 0.4=0.48\\P\left( W=2 \right) =0.6\cdot 0.6=0.36\\c:\\P\left( Z=0 \right) =0.4\\P\left( Z=1 \right) =0.6\\d:\\P\left( W\geqslant 1 \right) =1-P\left( W=0 \right) =1-0.16=0.84$