Explanation:
1)
the probability to roll a 4 is 1/6 (one desired case out of 6 possible cases).
the probability to roll something else is 1-1/6 = 5/6.
the probability of one possible combination of exactly 3 fours (and therefore 2 other numbers) is
(1/6)³×(5/6)²
this we need to multiply with the number of possible combinations (to pick 3 out of 5) :
C(5, 3) × (1/6)³×(5/6)² =
= (5! / (3! × (5-3)!)) × (1/6)³×(5/6)² = (5×4/2) × 25/(6)⁵ =
= 10 × 25/(6)⁵ = 250/7776 = 125/3888 = 0.032150206...
the probability to roll exactly 3 fours, when a die is rolled 5 times, is
0.032150206...
2)
the probability to get heads is 1/2 (one desired case out of 2 possible cases).
to get at least 5 heads in 6 attempts means the probability to get exactly 5 heads plus the probability to get 6 heads.
the probability to get 6 heads in 6 attempts is
1/2⁶ = 1/64
the probability to get exactly 5 heads (and 1 tail) is in one combination
(1/2)⁵× 1/2 = (1/2)⁶ = 1/64
and we have C(6, 5) possible combinations:
6! / (5! × (6-5)!) = 6
so, the probability to get exactly 5 heads in 6 attempts is
6 × 1/64 = 6/64
the probability to get at least 5 heads in 6 attempts is then
1/64 + 6/64 = 7/64 = 0.109375
3)
at most 1 head is the same as at least 5 tails, which is the same as at least 5 heads (because we have only 2 possible outcomes per toss).
so, it is the same as in 2) :
7/64 = 0.109375
4)
at most 2 heads is the probability to get at most 1 head plus the probability to get exactly 2 heads.
so, 3) + exactly 2 heads
to get exactly 2 heads (and 4 tails) in one combination is
(1/2)²×(1/2)⁴ = 1/2⁶ = 1/64
and we have C(6, 2) combinations :
6! / (2! × (6-2)!) = 6×5/2 = 3×5 = 15
so, the probability to get exactly 2 heads out of 6 attempts is
15 × 1/64 = 15/64
and so, the probability to get at most 2 heads is
7/64 + 15/64 = 22/64 = 11/32 = 0.34375
5)
to get at most 5 heads we need to add to 4) the probability to get exactly 3, exactly 4 and exactly 5 heads.
which is exactly the same as the opposite of getting 6 heads (every other result than 6 heads is desired).
the probability of the opposite of tossing 6 heads is
1 - probability to get 6 heads
the probability to get 6 heads is
1/2⁶ = 1/64
so,
1 - 1/64 = 63/64 = 0.984375
the probability to get at most 5 heads is
63/64 = 0.984375