1. From a bag containing 5 gold and 3 red marbles that feel identical, one is drawn at random. If the first marble drawn is not replaced before the second is drawn:
i) Explain why the probability of picking the second gold marble is 4/7 and why the probability of picking two gold marbles = 5/8 x 4/7.
ii) List all possibilities and their probabilities.
iii) If the experiment was repeated 560 times, how many times would you expect to get:
A) two gold marbles?
B) different coloured marbles?
2. A pencil case has 12 identically shaped coloured pencils in it. 3 are yellow, 4 are red and 5 are green. You pick two pencils one after another at random. (not replacing first, then picking another)
i) Explain why the probability of picking the second yellow pencil is 2/11, and why the probability of picking two yellow pencils is 3/12 x 2/11.
ii)What two coloured pencils are you most likely to pick?
iii) What two coloured pencils are you least likely to pick?
Explain your reasoning for both.
c) If 4 pencils are picked one after another at random, which of the following groups :
YRRG YYGG GGGG YRRR
do you expect to be:
i) more likely to be picked?
ii) least likely to be picked?
d) Work out the probability of each group above, and compare it with your expectations.
3. Caroline chooses a marble from a bag. Describe the possbile contents of this bag if the probability of her choosing :
a) a green marble is 1/8
b) a green marble is 1/8 the first time she chooses a marble, but 1/15 the second time, if the green marble was not put back or replaced after she picked it out the first time.
c) a green marble is 1/8 the first time she chooses a marble, but 3/31 the second time, if the green marble was not put back or replaced after she picked it out the first time.
4 a) Describe the possible contents of a bag if when Malcolm chooses a marble from the bag, the probability of him choosing a purple marble is 1/3.
b) If malcolm returns the purple marble to the bag in a), then chooses another marble, discuss what is the probability that the second marble is also purple.
c) If Malcolm keeps the purple marble in a) then chooses another, discuss what is the probability that the second marble is also purple.
d) If Malcolm chooses a marble from the bag, the probability of him choosing a black marble is 2/3 while the probability of him choosing a purple marble is 1/3. describe the posible contents of the bag.
THANK YOU VERY MUCH
We do not do your homework for you. However, I will give you a start.
i) The probability of picking a first gold marble is 5/8, since there are 5 gold marbles among the eight total. If you do not replace it, 4 gold marbles are among the remaining 7.
The probability of two or more events all occurring is found by multiplying the probabilities of the individual events.
I hope this helps.
1)
i) The probability of picking the second gold marble is 4/7 because after the first marble is drawn, there are now 4 gold marbles left out of a total of 7 marbles remaining in the bag. So, the probability is calculated as 4/7.
The probability of picking two gold marbles can be calculated by multiplying the probability of picking the first gold marble (5/8) by the probability of picking the second gold marble (4/7). This multiplication accounts for the fact that the second draw is dependent on the first draw and reflects the conditional probability. Therefore, the probability of picking two gold marbles is (5/8) x (4/7).
ii) Listing all possibilities and their probabilities:
- Two gold marbles: (5/8) x (4/7) = 20/56 = 5/14
- One gold and one red marble: (5/8) x (3/7) = 15/56
- One red and one gold marble: (3/8) x (5/7) = 15/56
- Two red marbles: (3/8) x (2/7) = 6/56 = 3/28
iii) If the experiment is repeated 560 times:
A) To calculate the expected number of times to get two gold marbles, multiply the probability of picking two gold marbles (5/14) by the number of repetitions: (5/14) x 560 = 200 times.
B) To calculate the expected number of times to get different colored marbles, subtract the probability of getting two gold marbles from 1 and multiply by the number of repetitions: (1 - 5/14) x 560 = 360 times.
2)
i) The probability of picking the second yellow pencil is 2/11 because after the first pencil is drawn without replacement, there are now 2 yellow pencils left out of a total of 11 pencils remaining in the pencil case. So, the probability is calculated as 2/11.
The probability of picking two yellow pencils can be calculated by multiplying the probability of picking the first yellow pencil (3/12) by the probability of picking the second yellow pencil (2/11). This multiplication accounts for the fact that the second draw is dependent on the first draw and reflects the conditional probability. Therefore, the probability of picking two yellow pencils is (3/12) x (2/11).
ii) The two colored pencils you are most likely to pick are the ones with the highest probabilities. In this case, yellow and red are the most likely since they have the highest numbers of pencils in the pencil case.
iii) The two colored pencils you are least likely to pick are the ones with the lowest probabilities. In this case, green-green (GG) would be the least likely, as green has the lowest number of pencils compared to yellow and red.
c) If 4 pencils are picked one after another at random:
i) The group YYGG is more likely to be picked since it has the highest probability, as calculated in part i) (3/12) x (2/11) x (4/10) x (3/9) = 1/55.
ii) The group GGGG is least likely to be picked since it has the lowest probability, as calculated in part i) (5/12) x (4/11) x (3/10) x (2/9) = 1/396.
3)
a) If the probability of Caroline choosing a green marble is 1/8, this means that out of the marbles in the bag, the number of green marbles divided by the total number of marbles is equal to 1/8. The possible contents of the bag could be 1 green marble and 7 non-green marbles, or any other combination that satisfies the 1/8 probability.
b) If the probability of picking a green marble changes to 1/15 after the first green marble is picked and not replaced, this means that the bag now has fewer green marbles. The possible contents of the bag could be 1 green marble and 6 non-green marbles, or any other combination that satisfies the new probability.
c) If the probability of picking a green marble changes to 3/31 after the first green marble is picked and not replaced, this means that the bag now has even fewer green marbles. The possible contents of the bag could be 1 green marble and 3 non-green marbles, or any other combination that satisfies the new probability.
4)
a) If when Malcolm chooses a marble from the bag, the probability of him choosing a purple marble is 1/3, this means that out of the marbles in the bag, the number of purple marbles divided by the total number of marbles is equal to 1/3. The possible contents of the bag could be 1 purple marble and 2 non-purple marbles, or any other combination that satisfies the 1/3 probability.
b) If Malcolm returns the purple marble to the bag in scenario a), then the probability of picking a purple marble on the second draw remains the same. Therefore, the probability of the second marble being purple is still 1/3.
c) If Malcolm keeps the purple marble in scenario a), then the probability of picking a purple marble on the second draw changes. Since the bag now has one less marble, the probability will depend on the new ratio of purple to non-purple marbles in the bag.
d) If Malcolm chooses a marble from the bag, and the probability of him choosing a black marble is 2/3 while the probability of him choosing a purple marble is 1/3, the possible contents of the bag could be 2 black marbles and 1 purple marble. However, without additional information about the total number of marbles in the bag, other possibilities satisfying the given probabilities cannot be determined.