An insurance sales representative selects 4 policies to review. The group of policies she can select from contains 10 life policies, 5 automobile policies, and 3 homeowners policies. Find the probability of selecting all life policies.
The number of ways to select all life insurance policies is 10C4. Hence,
P)all life policies = 10C4 = 210 = 7
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18C4 =3060 102
Can someone tell me how did you get the 210 and the 3060. I need it broke down step by step then I can follow it.
Thanks for your help. This is for my final.
To understand how to get the values of 210 and 3060 in this problem, let's break down the steps.
Step 1: Determine the total number of policies in the group.
In this case, there are 10 life policies, 5 automobile policies, and 3 homeowners policies. Therefore, the total number of policies available is 10 + 5 + 3 = 18.
Step 2: Determine the number of ways to select all life insurance policies.
To do this, we use the concept of combinations. In this problem, we want to select 4 policies from the 10 life policies available. The notation for this is denoted as 10C4, read as "10 choose 4".
The formula for combinations is: nCr = n! / (r!(n-r)!)
where n is the total number of items, r is the number of items you want to choose, and "!" denotes the factorial (e.g. 5! = 5 × 4 × 3 × 2 × 1).
So, calculating 10C4:
10C4 = 10! / (4!(10-4)!)
= 10! / (4!6!)
= (10 × 9 × 8 × 7 × 6!) / (4 × 3 × 2 × 1 × 6!)
= (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1)
= 210
Therefore, the number of ways to select all life insurance policies is 210.
Step 3: Determine the total number of ways to select any 4 policies from the group.
In this case, we want to calculate 18C4, which represents "18 choose 4".
Using the same combination formula as before:
18C4 = 18! / (4!(18-4)!)
= 18! / (4!14!)
= (18 × 17 × 16 × 15 × 14!) / (4 × 3 × 2 × 1 × 14!)
= (18 × 17 × 16 × 15) / (4 × 3 × 2 × 1)
= 3060
Therefore, the total number of ways to select any 4 policies is 3060.
Step 4: Calculate the probability of selecting all life insurance policies.
The probability is equal to the number of desired outcomes (selecting all life policies) divided by the total number of possible outcomes (selecting any 4 policies).
So, the probability of selecting all life insurance policies is:
P(all life policies) = 10C4 / 18C4
= 210 / 3060
≈ 0.0686 (rounded to 4 decimal places)
Therefore, the probability of selecting all life insurance policies is approximately 0.0686.
I hope this breakdown helps in understanding the steps and calculations involved. Good luck with your final!