A bag contains 3 red marbles, 5 yellow marbles and 6 orange marbles. Three marbles are drawn and NOT replaced each time. Determine the probability of drawing three yellow marbles in a row. State whether the events are dependent or independent.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
First marble = 5/14
With one less yellow marble, second = 4/13
With another less yellow marble, third = ?
Multiply the three values.
I'm still not sure.. I'm good with independent events but these dependent one has me confused..
I think I solved it:
RRR,YYYYY,OOOOOO
[RYO],[RRR],[RYY],[ROO]
[YYY],[YOR],[YOO],[YYO],[YRY],[YRR]
[OOO],[ORO],[OYO],[ORR],[OYY],[ORY]
There are 16 probable outcomes, only one of which was favourable, so 1/16=0.0625 so there is a 6% probability of you drawing 3 yellows!
This event is dependent as the probability of drawing another yellow marble can either reduce or increase depending on the amount of each colour that remains inside the bag.
To determine the probability of drawing three yellow marbles in a row, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Total number of marbles in the bag: 3 red + 5 yellow + 6 orange = 14 marbles
When we draw the marbles, we do not replace them back, so the number of marbles decreases with each draw.
To calculate the probability of drawing three yellow marbles in a row, we need to consider that the probability of the first yellow marble being drawn is 5/14 (since there are 5 yellow marbles out of a total of 14 marbles).
For the second draw, there are now 4 yellow marbles left out of 13 marbles.
Therefore, the probability of drawing the second yellow marble, given that the first was yellow, is 4/13.
For the third draw, there are 3 yellow marbles remaining out of 12 marbles.
So, the probability of drawing the third yellow marble, given that the first two were yellow, is 3/12 or 1/4.
To calculate the probability of all three events happening (drawing three yellow marbles in a row), we multiply the probabilities of each individual event since the events are independent: (5/14) * (4/13) * (1/4) = 20/728 or 5/182.
So, the probability of drawing three yellow marbles in a row is 5/182.
Now, to answer the second part of your question: Are the events dependent or independent?
Two events are said to be dependent if the outcome of one event affects the outcome of the other event. In this case, the marbles are not replaced after each draw, meaning that the number of yellow marbles changes with each draw. Therefore, the events are dependent since the probability of drawing a yellow marble is affected by the previous draws.