You are off to soccer and love being a goalkeeper, but that depends who is the coach today :
- with coach Ahmed the probablity of being a goalkeeper is 0.5
-with coach Usman the probability of being a goalkeeper is 0.3
Ahmed is coach more often.About 6 out of every 10 games ( a probability of 0.6)
a) Draw a probablity tree diagram to show all the outcomes of the experiment
b) what is the probablity that you will be a goalkeeper today
a) To draw a probability tree diagram, we can start by representing the different coaches as branches. Since the probability of being coached by Ahmed is 0.6, we will have a branch labeled "Ahmed" with a probability of 0.6. Similarly, we will have a branch labeled "Usman" with a probability of 0.4 (1 - 0.6).
Under each coach's branch, we will have two more branches representing the outcome of being a goalkeeper or not being a goalkeeper. Since the probability of being a goalkeeper with Ahmed is 0.5, we will have a branch labeled "Goalkeeper" with a probability of 0.5 under the "Ahmed" branch. Similarly, we will have a branch labeled "Not Goalkeeper" with a probability of 0.5 (1 - 0.5) under the "Ahmed" branch.
For coach Usman, the probability of being a goalkeeper is 0.3, so we will have a branch labeled "Goalkeeper" with a probability of 0.3 under the "Usman" branch. And under the "Usman" branch, we will have a branch labeled "Not Goalkeeper" with a probability of 0.7 (1 - 0.3).
The probability tree diagram will look like this:
Ahmed (0.6)
/ \
Goalkeeper (0.5) Not Goalkeeper (0.5)
Usman (0.4)
/ \
Goalkeeper (0.3) Not Goalkeeper (0.7)
b) To find the probability of being a goalkeeper today, we need to consider the probabilities of being a goalkeeper under each coach and the probability of each coach being present. We can calculate it using the law of total probability.
P(Goalkeeper today) = P(Goalkeeper | Ahmed) * P(Ahmed) + P(Goalkeeper | Usman) * P(Usman)
Substituting the values:
P(Goalkeeper today) = 0.5 * 0.6 + 0.3 * 0.4
= 0.3 + 0.12
= 0.42
Therefore, the probability of being a goalkeeper today is 0.42 (or 42%).