This is a classic probability question.
First you need to calculate the probability of each event. First add all the object together to get a total.
5 + 5 + 7 + 2 = 19
There are 5 engines, so 5/19 is the probability of drawing an engine
5+7 is the number of cars and so 12/19 is the probability of drawing a car
and there are 2 cabooses, so 2/19 is the probabilty of drawing a caboose.
Next, you assume that you DO draw the three objects that you wish. First, however, you must determine whether the events are dependent or independent. Since you will NOT be putting any cars back into the bag/box that you are drawing these piece out from, the events ARE dependent, as each draw will result in a different number of objects being in the box after each draw. (Said another way, when the outcome of one event affects the outcome of the second event, the two events are said to be dependent).
My source for the following quote is math planet:
"Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event. The probability of two dependent events is the product of the probability of X and the probability of Y AFTER X occurs.
P(XandY)=P(X)⋅P(Yafterx)"
In the case of our problem here, you will start by using the probability of drawing an engine for event #1 which is a probabilty of 5/19;
you will next be drawing for a car--but the probabilty we started with for this event was 12/19 which must be changed, because there are now only 18 objects in the box; this give us, therefore,
a probability of 12/18;
Last you will draw for a caboose--but again the probability has changed from the original probabilty we calculated which was 2/19 since now there are only 17 objects in the box.
Our probability is 2/17.
These three are multiplied together:
Fractions numerators and denomenators multiplied straight across in line, thus:
5 * 12 * 2
19 18 17
120
5,814
Now you reduce to lowest terms:
Probabilty of drawing these three cars in a row in three draws is:
20
969
I would love to work with you and your student--and offer a free demo lesson on how to solve this with an interactive online event for you. Contact me at: https://www.wyzant.com/Tutors/gentle_tutor
if you are interested.
