Richard has just been given a 8-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)
(a) What is the probability that he will answer all questions correctly?
(b) What is the probability that he will answer all questions incorrectly?
(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.
Then use the fact that P(r ≥ 1) = 1 − P(r = 0).
(d) What is the probability that Richard will answer at least half the questions correctly?

1. P(all are correct) =0.00000256

2. P ( all are incorrect) =  0.1677

3. P( at least one correct ) = 0.8310

c. P(r ≥ 1) = 0.8322

d. P ( half the questions are incorrect) = 0.046

Step-by-step explanation:

Let A be the event that he will choose the correct answer and there is only one correct answer out of 5 choices  so the probability of A = P(A)= 1/5

As there are 8 number of questions so n= 8

Applying the binomial theorem

P(all are correct) = ⁸C₈ (1/5)⁸ (4/5)°= 1* 1/5*5*5*5*5*5*5*5 (1)

a. P(all are correct) = 1/625 *625= 1/390625= 0.00000256

b. P ( all are incorrect) =  ⁸C₀ (1/5)° (4/5)⁸= 1* 4*4*4*4*4*4*4*4/5*5*5*5*5*5*5*5 (1)=  256*256/625 *625= 65336/390625 = 0.1677

c. P( at least one correct ) = P (one Correct) + P( 2 Correct) + P (3 correct) + P( 4 correct) + P ( 5 correct) + P ( 6  correct) + P ( 7  correct) + P ( 8  correct)

P( at least one correct ) = 0.00008192+  0.00114688+ 0.00917504 + 0.0458752 + 0.14680064 + 0.29360 +  0.335544 +0.00000256

P( at least one correct ) = 0.8310

P (one Correct) = ⁸C₇ (1/5)⁷ (4/5)= 8* 1/5*5*5*5*5*5*5  * 4/5 = 0.00008192

P( 2 Correct) = ⁸C₆ (1/5)⁶ (4/5)²= 28 * 1/5*5*5*5*5*5*  16/25=  0.00114688

P (3 correct)=⁸C5₅ (1/5)⁵ (4/5) ³= 56* 1/5*5*5*5*5*   64/125 = 0.00917504

P( 4 correct) = ⁸C₄(1/5)⁴ (4/5)⁴= 70* 1/5*5*5*5    * 256/625 = 0.0458752

P ( 5 correct) = ⁸C₃ (1/5)³ (4/5)⁵= 56 * 1/5*5*5    *1024/ 3125 = 0.14680064

P ( 6  correct)= ⁸C₂ (1/5)² (4/5) ⁶= 28* 1/5*5    * 4096 /15625= 0.29360

P ( 7  correct) = ⁸C₁ (1/5) (4/5)⁷=  8* 1/5   * 16384/78125 = 0.335544

P(r ≥ 1) = 1 − P(r = 0).

               =1- 0.1 677 = 0.8322

d. P ( half the questions are incorrect) =⁸C₄ (1/5)⁴ (4/5)⁴

= 70 * 1*1*1*1/ 5*5*5*5* 4*4*4*4/5*5*5*5 =  70 * 1/ 625* 256/625 = 0.0458= 0.046

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