30% of the 20 people in the Math Club have blonde hair. If 3 people are selected at random from the club, what is the probability that none have blonde hair?

Question

Grace Agate · Accepted Answer

#Pr=91/285#.

#30%# of 20 people is 6 people, so 6 people have blonde hair. This means that 14 people don't have blonde hair. If we let #a# equal non-blonde hair, and #b# equal blonde hair, that means #Pr(a)=7/10# and #Pr(b)=3/10#. If we think about it as picking the 3 people simultaneously, then the probability we seek is the chance of picking a group of 3 non-blondes. This is found by dividing the number of groups with 3 non-blondes #((14),(3))# by the total number of possible groups of 3 #((20),(3))#. With some simplification, we get #Pr("3 non-blondes")=(14!)/(11!" "3!)-:(20!)/(17!" "3!)# #=(14xx13xx12xxcancel(11!))/(cancel(11!)" "cancel(3!))-:(20xx19xx18xxcancel(17!))/(cancel(17!)" "cancel(3!))# #=(14xx13xx12)/(20xx19xx18)# #=(7xx13xx1)/(5xx19xx3)" "=" "91/285# If we think about it as picking the people one by one without replacing them, then the probability of a non-blonde getting picked the first time is #Pr(B_1)=7/10.# After a successful first pick, the probability for a non-blonde getting picked the second time is the number of non-blondes left (13) divided by the number of people left (19), which gives us #Pr(B_2|B_1)=13/19#. Finally, if we are successful on both the first and second picks, the probability of picking a non-blonde for a third time is, again, the number of non-blondes left (12) divided by the number of people left (18), which gives us #Pr(B_3|B_1, B_2)=12/18,# or #2/3#. To find the final probability, we multiply these three fractions together: #7/10*13/19*2/3# #=182/570# #=91/285# chance of picking 3 non-blondes one-by-one. What is the difference between independently occurring and mutually exclusive events? I hope I was of assistance.

Ezekiel Anderson · Answer

undefined To find the probability that none of the three people selected at random from the Math Club have blonde hair, we first find the probability that one person selected at random does not have blonde hair, which is $1 - 0.30 = 0.70$. Since the selection of each person is independent, we multiply the probabilities together for each person. So, the probability that none of the three people selected have blonde hair is $0.70 \times 0.70 \times 0.70 = 0.343$, or approximately $34.3\%$.