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Sagot :
Answer:
[tex]\textsf{a)} \quad \textsf{P(win)}=\dfrac{1}{658008}\approx 0.00000152 \; \sf (3\; s.f.)[/tex]
[tex]\textsf{b)} \quad \textsf{P(win)}=\dfrac{1}{1256640}\approx 0.000000796 \; \sf (3\; s.f.)[/tex]
c) See below.
Step-by-step explanation:
A person has bought one ticket each for two separate lotteries.
- For the first lottery, 5 random numbers are drawn from 40. To win the grand prize, we need to select the correct 5 numbers.
- For the second lottery, 4 random numbers are drawn one at a time from 35. To win the grand prize, we need to select the correct 4 numbers in the correct order.
To find the probability of winning the grand prize for each lottery, we can use the formula for probability:
[tex]\text{Probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}[/tex]
[tex]\dotfill[/tex]
Part a)
For the first lottery, 5 random numbers are drawn from 40. To win the grand prize, we need to select the correct 5 numbers.
The number of favorable outcomes is 1, as there is only one winning combination.
To determine the total number of outcomes, we need to use combinations without repetition, since we are selecting numbers from a larger group without repetition and without regard to the order in which they are chosen.
The number of ways to choose r items from a set of n items is:
[tex]\displaystyle \binom{n}{r}=\dfrac{n!}{r!(n-r)!}[/tex]
In this case r = 5 and n = 40, so:
[tex]\displaystyle \binom{40}{5}=\dfrac{40!}{5!(40-5)!} =658008[/tex]
Therefore, the probability of winning a grand prize where 5 random numbers are drawn from 40 is:
[tex]\textsf{Probability}=\dfrac{1}{658008}\approx 0.00000152 \; \sf (3\; s.f.)[/tex]
[tex]\dotfill[/tex]
Part b)
For the second lottery, 4 random numbers are drawn one at a time from 35. To win the grand prize, we need to select the correct 4 numbers in the correct order.
The number of favorable outcomes is 1, as there is only one winning combination.
To determine the total number of outcomes, we need to use permutations without repetition, since we are selecting numbers from a larger group without repetition and the order in which they are chosen matters.
The number of ways to choose r items from a set of n items, where no repetitions are allowed and order matters, is:
[tex]\dfrac{n!}{(n-r)!}[/tex]
In this case r = 4 and n = 35, so:
[tex]\dfrac{35!}{(35-4)!}=1256640[/tex]
Therefore, the probability of winning a grand prize where 4 random numbers are drawn one at a time from 35 and order matters is:
[tex]\textsf{Probability}=\dfrac{1}{1256640}\approx 0.000000796 \; \sf (3\; s.f.)[/tex]
[tex]\dotfill[/tex]
Part c)
The first lottery is more likely to be won even though it requires more correct numbers from a larger set.
This is because the second lottery requires the correct numbers to be selected in the correct order, significantly increasing the total number of possible outcomes, thereby decreasing the probability of winning.
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