Explanation:
a probability is always the ratio of
desired cases / totally possible cases
the probabilty to guess the correct answer to a multiple choice test is in our case
1/4 = 0.25
desired cases : 1 (the correct answer).
totally possible cases : the amount of answer options (4 in our case).
the probability to not get it right is then
3/4 = 0.75
desired cases : 3 (wrong answers in our case).
totally possible cases : 4 (in our case)
the probabilty of a combined event of independent individual events in parallel ("AND" relation) is the product of the individual probabilities.
the probabilty of a combined event of independent individual events as alternatives ("OR" relation) is the sum of the individual probabilities.
(a)
to get exactly 3 out of the 5 questions right means we have to multiply the probabilty to get the right answer 3 times by itself, and multiply that 2 times by the probabilty to get the wrong answer :
(1/4)³ × (3/4)² = 3²/4⁵ = 9/1024 = 0.0087890625
this is the probabilty for one of the possible constellations to get 3 right and 2 wrong (e.g. the first 3 are right, or the last 3 are right, or ...).
so, how many possible constellations are there to pick 3 out of 5 ?
these are the combinations without repetitions :
C(5, 3) = 5!/((5-3)! × 3!) = 5×4/2 = 5×2 = 10
so, we can have 10 different ways of having exactly 3 questions right out of 5.
each of the 10 has the same probabilty, and they are independent alternatives. so, we can simply add them up, which means we multiply the single probabilty by 10 :
to get exactly 3 questions right :
10 × 0.0087890625 = 0.087890625
to get exactly 4 questions right is (by using the same principles as before)
C(5, 4) × (1/4)⁴ × (3/4)¹ = 5 × 3/4⁵ = 15/1024 =
= 0.0146484375
the "or" relation of the question means we add the 2 probabilities :
10×9/1024 + 15/1024 = 105/1024 = 0.1025390625
(b)
the same principles apply as in (a).
to get exactly 2 wrong out of 5 is
C(5, 2) × (3/4)² × (1/4)³ = 10×9/1024 = 0.087890625
to get exactly 3 wrong out of 5 is
C(5, 3) × (3/4)³ × (1/4)² = 10×27/1024 = 0.263671875
and again, for the combined "or" probabilty we add both :
10×9/1024 + 10×27/1024 = 360/1024 = 0.3515625
(c)
that means she gets exactly 1 or exactly 2 or exactly 3 or exactly 4 or exactly 5 questions right.
or, in other words, she does NOT get all 5 wrong.
that means the requested probabilty is complementary to the probabilty of getting all 5 wrong :
1 - P(getting all 5 wrong)
as you can see it is much easier to calculate the probability for having all 5 wrong and then subtract it from the sure event (1), than calculating all the individual events and sum them up.
to get all 5 wrong is simply
C(5, 5) × (3/4)⁵ = 1 × 243/1024 = 0.2373046875
so, the probabilty to get at least 1 question right is
1 - 0.2373046875 = 0.7626953125
(d)
in a similar way of thinking this is complementary to getting either no question wrong or getting exactly 1 question wrong :
1 - P(no wrong) - P(exactly 1 wrong)
to get no question wrong (= all correct) :
C(5, 0) × (3/4)⁰ × (1/4)⁵ = 1×1 × 1/1024 = 0.0009765625
to get exactly 1 question wrong :
C(5, 1) × (3/4)¹ × (1/4)⁴ = 5×3/1024 = 15/1024 =
= 0.0146484375
the probability to get at least 2 questions wrong is then
1 - 1/1024 - 15/1024 = 1 - 16/1024 = 1008/1024 =
= 0.984375