Two people are selected at random from a group of 6 pilots and 4 engineers. What is the probability that both of them are engineers?
Homework Posting Tips
Please be patient. All tutors are volunteers, and sometimes a tutor may not be immediately available. Please be patient while waiting for a response to your question.
To calculate the probability that both selected individuals are engineers, we need to find the ratio of the number of favorable outcomes (both engineers) to the total number of possible outcomes.
Step 1: Calculate the total number of possible outcomes.
In a group of 6 pilots and 4 engineers, there are a total of 10 individuals. Therefore, the total number of possible outcomes is given by selecting any two individuals from this group, which is represented by the combination formula (nCr).
Total number of possible outcomes = 10C2 = (10!)/(2!(10-2)!) = (10!)/(2!8!) = (10*9)/(2*1) = 90/2 = 45
Step 2: Calculate the number of favorable outcomes.
We want to select two engineers from the group of 4 engineers. Since order doesn't matter, we can use the combination formula again.
Number of favorable outcomes = 4C2 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4*3)/(2*1) = 6/2 = 3
Step 3: Calculate the probability.
The probability is given by the ratio of favorable outcomes to total outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3/45
Step 4: Simplify the probability, if necessary.
In this case, the probability cannot be simplified further as there are no common factors between 3 and 45.
Therefore, the probability that both selected individuals are engineers is 3/45, which can also be simplified to 1/15.
To find the probability that both selected individuals are engineers, we need to calculate the ratio between the favorable outcomes (both selected individuals are engineers) and the total number of possible outcomes.
Step 1: Determine the number of ways to select 2 individuals from the total group of pilots and engineers.
This can be done using the combination formula, which calculates the number of ways to choose k items from a set of n items. In this case, we want to select 2 individuals from a group of 10 (6 pilots and 4 engineers). The formula is:
C(n, k) = n! / (k!(n - k)!)
C(10, 2) = 10! / (2!(10 - 2)!)
= 10! / (2!8!)
= (10 * 9 * 8!) / (2!8!)
= (10 * 9) / 2
= 45
So, there are 45 possible ways to select 2 individuals from the group of pilots and engineers.
Step 2: Determine the number of ways to select 2 engineers from the group of 4 engineers.
To calculate this, we use the combination formula again:
C(4, 2) = 4! / (2!(4 - 2)!)
= 4! / (2!2!)
= (4 * 3 * 2!) / (2!2!)
= (4 * 3) / 2
= 6
There are 6 possible ways to select 2 engineers from the group of 4 engineers.
Step 3: Calculate the probability.
The probability of both selected individuals being engineers is given by the ratio of the number of favorable outcomes (6) to the total number of possible outcomes (45).
Probability = favorable outcomes / total outcomes
= 6 / 45
= 2 / 15
Therefore, the probability that both individuals selected are engineers is 2/15 or approximately 0.1333.
number of choices of any 2 of the 10
= C(10,2) = 45
number of choices of any 2 of the 4 engineers
= C(4,2) = 6
Prob(of stated event) = 6/45 = 2/15
or, simply
prob(of stated event) = (4/10)(3/9) = 2/15