Question

Write the probability of each italicized event in symbols (e.g., P(X ≥ 5)). (a) At least 5 correct answers on a 16-question quiz (X = number of correct answers). (b) Fewer than 3 “phishing” e-mails out of 25 e-mails (X = number of phishing e-mails). (c) At most 7 no-shows at a party where 19 guests were invited (X = number of no-shows).

Answer 1

(a) Assuming each question has just 2 options- right or wrong, P(X ≥ 5) = C(16, 5) (1/2)^5 (1/2)^(16 - 5)

P(X ≥ 5) = C(16, 5) (1/2)^5 (1/2)^11

(b) Assuming each email is either a phishing email or non-phishing email, P(X < 3) = P(0) + P(1) + P(2)

= C(25, 0) (1/2)^0 (1/2)^(25 - 0) + C(25, 1) (1/2)^1 (1/2)^(25 - 1) + C(25, 2) (1/2)^2 (1/2)^(25 - 2)

P(X < 3) = C(25, 0) (1/2)^0 (1/2)^25 + C(25, 1) (1/2)^1 (1/2)^24 + C(25, 2) (1/2)^2 (1/2)^23

(c) Assuming just two possibilities- show or no show, P(X ≤ 7) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7)

= C(19, 0) (1/2)^0 (1/2)^(19 - 0) + C(19, 1) (1/2)^1 (1/2)^(19 - 1) + C(19, 2) (1/2)^2 (1/2)^(19 - 2) + C(19, 3) (1/2)^3 (1/2)^(19 - 3) + C(19, 4) (1/2)^4 (1/2)^(19 - 4) + C(19, 5) (1/2)^5 (1/2)^(19 - 5) + C(19, 6) (1/2)^6 (1/2)^(19 - 6) + C(19, 7) (1/2)^7 (1/2)^(19 - 7)

P(X ≤ 7) = C(19, 0) (1/2)^0 (1/2)^19 + C(19, 1) (1/2)^1 (1/2)^18 + C(19, 2) (1/2)^2(1/2)^17 + C(19, 3) (1/2)^3 (1/2)^16
+ C(19, 4) (1/2)^4 (1/2)^15 + C(19, 5) (1/2)^5 (1/2)^14 + C(19, 6) (1/2)^6 (1/2)^13 + C(19, 7) (1/2)^7 (1/2)^12