Coin toss - probability My Question is -
Each round Mike and Dean toss coin each. Mike tosses a not fair coin in which the probability to get heads is $0.6$. Dean tosses a not fair coin in which the probability to get heads is $0.1$. they toss the coins till they get the same results at the same time. 
What is the probability that there will be at most 5 rounds?
I started to calculate it as geometric distribution but something doesn't seem right in my calculations. I thought so since they are throwing till 'success" which defined Geometric probability. 
 A: The probability till Mike and Dean get the same results at most 5 rounds:
The final results can be... $${HH}\ or\ {TT}$$
The probability that Mike and Dean get the same results in a round
$$={0.6}\times{0.1}+{(1-0.6)}\times{(1-0.1)}=0.42 $$
The probability that Mike and Dean do not get the same results in a round $$={1-0.42}=0.58$$
Let X be the number of rounds until Mike and Dean get the same results. $X\sim Geo(0.42)$
$$P(X\leq5)=P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)$$
$$P(X\leq5)=0.42+0.58\times0.42+(0.58)^2\times0.42+(0.58)^3\times0.42+(0.58)^4\times0.42\approx0.9344\ (corr.to\ 4\ d.p.)$$
A: Mark andDean have 5 rounds in which they need at least one successful toss. P(successful toss)=0.6*0.1=0.01
Next we use 1-(1-p)^n
1-(1-0.06)^5=0.2661
A: The game ends when Mike and Dean toss different results.


*

*Mike: Heads=0.6 + Tails=0.4

*Dean: Heads=0.1 + Tails=0.9


The probability of round being:


*

*Both Heads = 0.60 * 0.40 = 0.24

*Both Tails = 0.10 * 0.90 = 0.09

*Both Same  = 0.24 + 0.09 = 0.33

*Both Different = 1 - 0.33 = 0.67


What is the probability that of getting to round X?


*

*1 = 100%    (first round is always played)

*2 = 0.33    (both same)

*3 = 0.33^2 = 0.1089  (both same, twice)

*4 = 0.33^3 = 0.0359  (both same, 3x)

*5 = 0.33^4 = 0.0118  (both same, 4x)

*6 = 0.33^5 = 0.0039  (both same, 5x)


What is the probability that there will be at most 5 rounds? 


*

*This is simply 1 minus the probability of getting to round 6

*(1 - 0.33^5) = 0.99608 

