Question

1. to win a certain lottery game, a player must correctly choose 4 numbers from the numbers 1 - 25. numbers can not repeat. what is the probability of winning this lottery?
2. jaylon is deciding on what order to visit france, italy, spain, england, germany, and ireland within the next two years. if he randomly chooses the order, what is the probability he visits spain first?
3. three letters are chosen at random from the word transformation. what is the probability that they are all vowels?
4. there are 28 girls and 21 boys in chorus. if two are chosen at random to sing a duet, what is the probability that at least one is a girl?
5. there are nine golf balls numbered 1 - 9 in a bag. five of the balls are selected at random to create a 4 - digit number. what is the probability that the number is at least 3,000?

Explanation:

Step1: Calculate total combinations for lottery

Use combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 25$, $r=4$. So $C(25,4)=\frac{25!}{4!(25 - 4)!}=\frac{25\times24\times23\times22}{4\times3\times2\times1}=12650$. Probability of winning is $\frac{1}{12650}$.

Step2: Calculate probability for Jaylon

There are 5 countries, probability of Spain first is $\frac{1}{5}$.

Step3: Count letters in TRANSFORMATION

There are 13 letters, 5 vowels. $C(13,3)=\frac{13!}{3!(13 - 3)!}=286$, $C(5,3)=\frac{5!}{3!(5 - 3)!}=10$. Probability is $\frac{10}{286}=\frac{5}{143}$.

Step4: Calculate probability for chorus

Total people: $28 + 21=49$. Probability of no - girls (2 boys) is $\frac{C(21,2)}{C(49,2)}=\frac{\frac{21!}{2!(21 - 2)!}}{\frac{49!}{2!(49 - 2)!}}=\frac{21\times20}{49\times48}=\frac{5}{28}$. Probability of at least 1 girl is $1-\frac{5}{28}=\frac{23}{28}$.

Step5: Calculate probability for golf balls

For 4 - digit number $\geq3000$, first digit must be 3 - 9. Total ways to pick 4 balls out of 5 for number: $A(9,4)=\frac{9!}{(9 - 4)!}=3024$. Ways with first digit 1 or 2: $2\times A(8,3)=\frac{2\times8!}{(8 - 3)!}=672$. Probability is $1-\frac{672}{3024}=\frac{3024 - 672}{3024}=\frac{2352}{3024}=\frac{4}{6}=\frac{2}{3}$.

Answer:

1. $\frac{1}{12650}$
2. $\frac{1}{5}$
3. $\frac{5}{143}$
4. $\frac{23}{28}$
5. $\frac{2}{3}$