Question #9ec4b
1 Answer
Explanation:
Finding the probability of at least
The probability of
#2# people out of#5# having a dog is equivalent to the probability of#2# people having a dog and#3# people not having a dog.
#overbrace(3/5 * 3/5) ^ "dog" * overbrace(2/5 * 2/5 * 2/5) ^ "no dog"# However, we don't know which two people own dogs. It can be the first and second person, first and fifth person, fourth and fifth person, etc. Basically, it can be any set of
#2# people out of the#5# . Thus, we have to multiply the combination#color(white)(I)_5C_2# along with everything.A quick recap on combinations:
#color(white)(I)_5C_2 = ((5), (2)) = (5!)/(2! * (5-2)!) = 10# You can also plug
#color(white)(I)_5C_2# into your calculator, which is what I'll be doing below.So, the probability of
#2# people owning a dog is actually
#(3/5)^2 * (2/5)^3 * color(white)(I)_5C_2 = color(red)0.2304#
We can do the same thing for all the other possibilities.
Probability of
#3# people owning a dog:
#(3/5)^3 * (2/5)^2 * color(white)(I)_5C_3 = color(red)0.3456# Probability of
#4# people owning a dog:
#(3/5)^4 * (2/5)^1 * color(white)(I)_5C_4 = color(red)0.2592# Probability of
#5# people owning a dog:
#(3/5)^5 * (2/5)^0 * color(white)(I)_5C_5 = color(red)0.07776#
Now that we have found each of the individual probabilities, we can add them up:
#color(red)0.2304 + color(red)0.3456 + color(red)0.2592 + color(red)0.07776 = 0.91296#
Again, this is the probability that
#0.91296 ~~ color(blue)(91.3%)#
