Answer:
a) 6
b) 4
c) 3
Step-by-step explanation:
Let p be the probability of having a female Martian, and of course, 1-p the probability of having a male Martian.
To compute the expected total number of trials before 2 males are born, imagine an experiment simulating the fact that 2 males are born is performed n times.
Let ak be the number of trials performed until 2 males are born in experiment k. That is,
a1= number of trials performed until 2 males are born in experiment 1
a2= number of trials performed until 2 males are born in experiment 2
and so on.
If a1 + a2 + … + an = N
we would expect Np females.
Since the experiment was performed n times, there 2n males (recall that the experiment stops when 2 males are born).
So we would expect 2n = N(1-p), or
N/n = 2/(1-p)
But N/n is the average number of trials per experiment, that is, the expectation.
<em>We have then that the expected number of trials before 2 males are born is 2/(1-p) where p is the probability of having a female.
</em>
a)
Here we have the probability of having a male is half as likely as females. So
1-p = p/2 hence p=2/3
The expected number of trials would be
2/(1-2/3) = 2/(1/3) =6
This means <em>the couple would have 6 children</em>: 4 females (the first 4 trials) and 2 males (the last 2 trials).
b)
Here the probability of having a female = probability of having a male = 1/2
The expected number of trials would be
2/(1/2) = 4
This means<em> the couple would have 4 children</em>: 2 females (the first 2 trials) and 2 males (the last 2 trials).
c)
Here, 1-p = 2p so p=1/3
The expected number of trials would be
2/(1-1/3) = 2/(2/3) = 6/2 =3
This means<em> the couple would have 3 children</em>: 1 female (the first trial) and 2 males (the last 2 trials).
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