Q:

Stan guessed on all 10 questions of a multiple-choice quiz. Each question has 4 answer choices. What is the probability that he got at least 2 questions correct? Round the answer to the nearest thousandth.

Accepted Solution

A:
The probability that he got at least 2 questions correct is 0.756Lets assume, the event of getting a question correct is P As, each question has 4 answer choices and only one choice is correct, So, P = [tex] \frac{1}{4} [/tex]When
the probability of getting 1 success in 1 trial is p, then the probability of getting exactly x successes out of n trials is given by the
formula : (ⁿCₓ )(P)ˣ (1-P)ⁿ⁻ˣ Here the total number of questions is 10. So, n= 10Stan needs to get at least 2 questions correct, that means he can get 2, 3, 4, 5, 6, 7, 8, 9 or 10 questions correct.Now according to the formula, P(x=2) = ¹⁰C₂ ([tex] \frac{1}{4} [/tex])² ([tex] 1- \frac{1}{4} [/tex])¹⁰⁻² = ¹⁰C₂ [tex] (\frac{1}{16})(\frac{3}{4})^8 [/tex]= 0.281567573 P(x=3) = ¹⁰C₃ ([tex] \frac{1}{4} [/tex])³ [tex] (\frac{3}{4})^7 [/tex]= (120) ([tex] \frac{1}{4} [/tex])³ [tex] (\frac{3}{4})^7 [/tex]= 0.250282287P(x=4) = ¹⁰C₄ ([tex] \frac{1}{4} [/tex])⁴ [tex] (\frac{3}{4})^6 [/tex]= (210)([tex] \frac{1}{4} [/tex])⁴ [tex] (\frac{3}{4})^6 [/tex]= 0.145998001P(x=5) = ¹⁰C₅ ([tex] \frac{1}{4} [/tex])⁵ [tex] (\frac{3}{4})^5 [/tex]= (252) ([tex] \frac{1}{4} [/tex])⁵ [tex] (\frac{3}{4})^5 [/tex]= 0.058399200P(x=6) = ¹⁰C₆ ([tex] \frac{1}{4} [/tex])⁶ [tex] (\frac{3}{4})^4 [/tex]= (210) ([tex] \frac{1}{4} [/tex])⁶ [tex] (\frac{3}{4})^4 [/tex]= 0.016222000P(x=7) = ¹⁰C₇ ([tex] \frac{1}{4} [/tex])⁷ [tex] (\frac{3}{4})^3 [/tex]= (120) ([tex] \frac{1}{4} [/tex])⁷ [tex] (\frac{3}{4})^3 [/tex]= 0.003089904P(x=8) = ¹⁰C₈ ([tex] \frac{1}{4} [/tex])⁸ [tex] (\frac{3}{4})^2 [/tex]= (45) ([tex] \frac{1}{4} [/tex])⁸ [tex] (\frac{3}{4})^2 [/tex]= 0.000386238P(x=9) = ¹⁰C₉ ([tex] \frac{1}{4} [/tex])⁹ [tex] (\frac{3}{4})^1 [/tex]= (10) ([tex] \frac{1}{4} [/tex])⁹ [tex] (\frac{3}{4})^1 [/tex]= 0.000028610 P(x=10) = ¹⁰C₁₀ ([tex] \frac{1}{4} [/tex])¹⁰ = 0.000000953Now we will just add all the probabilities and get: 0.281567573 + 0.250282287+ 0.145998001+ 0.058399200+ 0.016222000+0.003089904+0.000386238+ 0.000028610 +0.000000953 = 0.755974766 = 0.756 (rounding to the nearest thousandth)So, the probability that he got at least 2 questions correct is 0.756