SOLUTION MANUAL FOR A FIRST COURSE IN
PROBABILITY 2026 FINAL PAPER COMPLETE
REVISION QUESTIONS WITH FULL SOLUTIONS
◉ What is the probability of drawing a specific card from a deck?
Answer: P(specific card) = 1/52.
◉ What is the probability of drawing a specific color from a bag of
colored balls? Answer: P(specific color) = number of specific color
balls/total balls.
◉ What is a 3-tuple in probability? Answer: A 3-tuple is an ordered
triple of outcomes.
◉ What is the probability of getting a combination of outcomes?
Answer: Add the probabilities of all favorable combinations.
◉ What is the formula for combinations? Answer: nCr = n! / (r!(n-
r)!).
◉ What is the probability of drawing 2 green balls and 1 yellow from
a set of 3 green and 4 yellow? Answer: P(2 greens and one yellow) =
120/1001.
,◉ How many ways can you choose 4 distinct elements from a set of
14? Answer: 14 · 13 · 12 · 11 = 24024.
◉ What is the probability of hitting within 2 inches of the center of a
dartboard with a 20-inch side length? Answer: P(A) = π·2² / 20² =
π/100.
◉ What is the probability of needing at most 3 rolls to see the
number four on a die? Answer: P(need at most 3 rolls) = 91/216.
◉ What is the probability that one of the colors does not appear
when drawing 3 balls from a set of 2 red, 3 green, and 3 white?
Answer: P(W [ G [ R) = 13/16.
◉ What is the probability of the event A where a stick is broken at a
point uniformly chosen along its length L? Answer: P(A) = L/5 + L/5
= 2/5.
◉ What is the probability of the number four never appearing when
rolling a die? Answer: P(the number four never appears) = 0.
◉ What is the probability of choosing a student who wears a watch
or a bracelet? Answer: P(W [ B) = 0.4.
, ◉ What is the probability of having an even number of rolls when
rolling a die until a four appears? Answer: P(even number of rolls) =
5/11.
◉ How many ways can you choose 2 green balls from 3 green and 1
yellow from 4 yellow? Answer: P(2 greens and one yellow) = (3
choose 2) * (4 choose 1) = 120.
◉ What is the probability of drawing 4 balls from a set of 2 red, 3
green, and 3 white? Answer: P(W [ G [ R) = 13/43.
◉ What is the total number of outcomes when flipping a coin 5
times? Answer: #⌦ = 2^5 = 32.
◉ What is the probability of drawing 4 distinct elements from a set
of 14? Answer: P(4 distinct) = 24024.
◉ What is the probability of drawing 2 numbers between 1 and 5, 1
between 6 and 9, and 1 between 10 and 12? Answer: P(C) =
120/1001.
◉ What is the probability of needing k rolls to see the first four?
Answer: P(k) = (5/6)^(k-1) * (1/6).
PROBABILITY 2026 FINAL PAPER COMPLETE
REVISION QUESTIONS WITH FULL SOLUTIONS
◉ What is the probability of drawing a specific card from a deck?
Answer: P(specific card) = 1/52.
◉ What is the probability of drawing a specific color from a bag of
colored balls? Answer: P(specific color) = number of specific color
balls/total balls.
◉ What is a 3-tuple in probability? Answer: A 3-tuple is an ordered
triple of outcomes.
◉ What is the probability of getting a combination of outcomes?
Answer: Add the probabilities of all favorable combinations.
◉ What is the formula for combinations? Answer: nCr = n! / (r!(n-
r)!).
◉ What is the probability of drawing 2 green balls and 1 yellow from
a set of 3 green and 4 yellow? Answer: P(2 greens and one yellow) =
120/1001.
,◉ How many ways can you choose 4 distinct elements from a set of
14? Answer: 14 · 13 · 12 · 11 = 24024.
◉ What is the probability of hitting within 2 inches of the center of a
dartboard with a 20-inch side length? Answer: P(A) = π·2² / 20² =
π/100.
◉ What is the probability of needing at most 3 rolls to see the
number four on a die? Answer: P(need at most 3 rolls) = 91/216.
◉ What is the probability that one of the colors does not appear
when drawing 3 balls from a set of 2 red, 3 green, and 3 white?
Answer: P(W [ G [ R) = 13/16.
◉ What is the probability of the event A where a stick is broken at a
point uniformly chosen along its length L? Answer: P(A) = L/5 + L/5
= 2/5.
◉ What is the probability of the number four never appearing when
rolling a die? Answer: P(the number four never appears) = 0.
◉ What is the probability of choosing a student who wears a watch
or a bracelet? Answer: P(W [ B) = 0.4.
, ◉ What is the probability of having an even number of rolls when
rolling a die until a four appears? Answer: P(even number of rolls) =
5/11.
◉ How many ways can you choose 2 green balls from 3 green and 1
yellow from 4 yellow? Answer: P(2 greens and one yellow) = (3
choose 2) * (4 choose 1) = 120.
◉ What is the probability of drawing 4 balls from a set of 2 red, 3
green, and 3 white? Answer: P(W [ G [ R) = 13/43.
◉ What is the total number of outcomes when flipping a coin 5
times? Answer: #⌦ = 2^5 = 32.
◉ What is the probability of drawing 4 distinct elements from a set
of 14? Answer: P(4 distinct) = 24024.
◉ What is the probability of drawing 2 numbers between 1 and 5, 1
between 6 and 9, and 1 between 10 and 12? Answer: P(C) =
120/1001.
◉ What is the probability of needing k rolls to see the first four?
Answer: P(k) = (5/6)^(k-1) * (1/6).
