How to solve probability problems
Level 1
A hotel has 2 rooms by the seashore and 4 rooms by the golf course vacant.
If you check in, what are the chances you'll get a room by the seashore? 1/3
Solution
- Rooms by the seashore: 2
- Total number of rooms available: 6
- Probability of receiving a room by the seashore: 2/6 = 1/3
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Level 2
On Monday and Wednesday night,
Mrs. Miller went to bed listening to her favorite music CD.
The CD contains music from 2 different composers:
4 tracks are Scarlatti and 4 tracks are Debussy.
Her stereo is programmed to play the tracks in a random order.
How likely is it that on both nights the last track played was Scarlatti? 1/4
Solution
- Probability that Monday night the last track was Scarlatti: 4/8
- Probability that Wednesday night the last track was Scarlatti: 4/8
- Probability of finishing with Scarlatti both Monday and Wednesday: 4/8 × 4/8 = 1/4
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Level 3
Angelita has some marbles in two bags.
In one bag she has 3 orange, 3 black and 3 yellow marbles.
In the other bag she has 3 orange, 4 black and 5 yellow marbles.
If Angelita takes one marble out of each bag,
how likely is it that neither one is orange? 1/2
Solution
- Calculate the probability that both bags yielded black or yellow marbles.
- Probability that bag one yielded a black or yellow marble: 6/9
- Probability that bag two yielded a black or yellow marble: 9/12
- Probability that Angelita did not draw an orange marble: 6/9 × 9/12 = 1/2
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Level 4
Alejandro has some candies in his pockets.
In his left pocket he has 1 chocolate and 4 lollipops.
In his right pocket he has 3 chocolates and 1 lollipop.
If Alejandro takes one candy out of each pocket,
what is the probability that at least one of the candies is a chocolate? 4/5
Solution
- Calculate the probability that neither candy was a chocolate (both were lollipops), and subtract this fraction from 1.
- Probability that he drew a lollipop from his left pocket: 4/5
- Probability that he drew a lollipop from his right pocket: 1/4
- Probability that both candies were lollipops (neither was a chocolate): 4/5 × 1/4 = 1/5
- Probability that at least one candy was a chocolate: 1 - 1/5 = 4/5
Solution 2
- Calculate all cases.
- Left pocket chocolate and right pocket chocolate: 1/5 × 3/4 = 3/20
- Left pocket chocolate and right pocket lollipop: 1/5 × 1/4 = 1/20
- Left pocket lollipop and right pocket chocolate: 4/5 × 3/4 = 12/20
- Left pocket lollipop and right pocket lollipop: 4/5 × 1/4 = 4/20
- Cases 1-3 above are successes for drawing a chocolate.
- Probability that at least one candy was a chocolate: 3/20 + 1/20 + 12/20 = 16/20 = 4/5
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Level 5
In the morning Angelita goes fishing at Wide Pond,
and in the afternoon at Shallow Pond.
The fish in Wide Pond are 1 carp and 2 sunfish,
and in Shallow Pond they are 2 carp and 6 sunfish.
If Angelita catches one fish in each pond,
how likely is it that both are the same kind of fish? 7/12
Solution
- Add the probability that both fish are carp and the probability that both are sunfish.
- Probability that the fish from Wide Pond is a carp: 1/3
- Probability that the fish from Wide Pond is a sunfish: 2/3
- Probability that the fish from Shallow Pond is a carp: 2/8
- Probability that the fish from Shallow Pond is a sunfish: 6/8
- Probability that both fish are carp: 1/3 × 2/8 = 1/12
- Probability that both fish are sunfish: 2/3 × 6/8 = 1/2
- Probability that both are carp OR both are sunfish: 1/12 + 1/2 = 7/12
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