A space station has 5 cosmonauts on board at all times, of whom 3 are experienced, 2
inexperienced. Occasionally some work must be done outside the space station. The
probability of being selected to carry out the work is twice as large for each individual
experienced cosmonaut as it is for each inexperienced cosmonaut. Experienced cosmonauts have a 10% chance of getting into difficulties whilst outside, inexperienced
cosmonauts 25%.
(i) Calculate the probability that the cosmonaut selected is an experienced one.
To calculate the probability that the cosmonaut selected is an experienced one, we need to consider the number of experienced and inexperienced cosmonauts on board and the probabilities of being selected for the work.
Given that there are 3 experienced and 2 inexperienced cosmonauts, the total number of cosmonauts on board is 5.
Let's represent the probability of being selected for the work as:
P(experienced) = probability of an experienced cosmonaut being selected
P(inexperienced) = probability of an inexperienced cosmonaut being selected
According to the problem, the probability of an experienced cosmonaut being selected is twice as large as the probability of an inexperienced cosmonaut being selected.
Let's say the probability of an inexperienced cosmonaut being selected is x. Therefore, the probability of an experienced cosmonaut being selected is 2x.
Since the total probabilities must sum up to 1, we can set up the equation:
P(experienced) + P(inexperienced) = 1
Substituting the values:
2x + x = 1
3x = 1
x = 1/3
Now we can calculate the probability of an experienced cosmonaut being selected:
P(experienced) = 2x = 2(1/3) = 2/3
Therefore, the probability that the cosmonaut selected is an experienced one is 2/3 or approximately 0.667.