Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. A 4-digit number greater than 5000 is randomly formed from the digits 0, 1, 3, 5, 7 (repetition allowed). What is the probability of forming a number divisible by 5?
2. If 7 relatives (3 ladies and 4 gentlemen) are to be invited for dinner, find the probability that at least one lady is included in a selection of 3 guests.
3. Two dice are thrown simultaneously. The probability that the product of the numbers appearing is even is
4. If A and B are mutually exclusive events such that $P(A) = 0.35$ and $P(B) = 0.45$, then $P(A' \cap B')$ is
5. A 5-digit number is formed using the digits 0, 1, 2, 3, 4 without repetition. The probability that it is divisible by 4 is
6. Four cards are drawn from a well-shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and one spade?
7. The letters of the word 'SOCIETY' are placed at random in a row. What is the probability that the three vowels come together?
8. Three numbers are chosen from the set ${1, 2, \dots, 20}$. Find the probability that they are in A.P.
9. An integer is chosen at random from the first 200 positive integers. What is the probability that the integer is divisible by 6 or 8?
10. 6 boys and 6 girls sit in a row at random. Find the probability that (i) all the girls sit together, (ii) the boys and girls sit alternately.
11. If $E$ and $F$ are events such that $P(E) = 1/4, P(F) = 1/2$ and $P(E \cap F) = 1/8$. Find:
    (i) $P(E \text{ or } F)$
    (ii) $P(\text{not } E \text{ and not } F)$
    (iii) $P(\text{not } E)$
    (iv) Are E and F independent?
    (v) Are E and F mutually exclusive?
12. Case Study: A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has:
    (i) no girl?
    (ii) at least one boy and one girl?
    (iii) at least 3 girls?
    Find the probability for each case if a team of 5 is selected at random.