问题1.一枚硬币被扔一次。写下其样本空间。
解决方案：
Since, a coin has two faces either head or tails. So, if the coin is tossed it will land on either of these two sides. Hence, sample space will be given as:
S = { H, T }, where H is the event Head comes upon landing and T is the event Tails comes upon landing.
问题2.如果将硬币扔了两次，请描述与此实验相关的样本空间。
解决方案：
If a coin is tossed two times, then there can be four possible cases which are:
(A) both coins show head
(B) first coin shows head and second shows tail
(C) first coin shows tail and second shows head
(A) Both coins show tail
Hence, sample space will be given as:
S = { HH, HT, TH, TT }
问题3.如果将一枚硬币扔三遍(或将三枚硬币扔在一起)，请描述该实验的样本空间。
解决方案：
If a coin is tossed three times (or three coins are tossed together), then there will be 2^3 = 8 possible cases which are:
(A) all coins show head
(B) first two coins show head and third shows tail
(C) first coin shows head and other two shows tail
(D) first coin shows tail and other two shows head
(E) first coin shows head second shows tail and third shows head
(F) first coin shows tail second shows head and third shows tail
(G) first two coins show tail and third shows head
(H) all coins show tail
Hence, sample space will be given as:
S = { HHH, HHT, HTT, THH, HTH, THT, TTH, TTT }
问题4.编写样本空间以进行四次抛硬币实验。
解决方案：
If a coin is tossed four times, then there will be 2^4 = 16 possible cases and its sample space will be given as:
S = { HHHH, HHHT, HHTT, HHTH, HTHH, THHH, HTTH, HTHT,
TTTT, TTTH, TTHH, TTHT, THTT, HTTT, THHT, THTH }
问题5.掷两个骰子。描述此实验的样本空间。
解决方案：
Since, a dice has six faces, when a dice is thrown two times, there can be 6^2 = 36 possible cases, so the sample space can be written as:
S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
问题6.与将三个骰子放在一起的随机实验相关的基本事件总数是多少?
解决方案：
Since, a dice has six faces, when a dice is thrown three times, total number of elementary events associated will be 6^3 = 216 events.
问题7.投掷硬币，然后掷出骰子。描述此实验的样本空间。
解决方案：
When a coin is tossed it can land on either of two sides Head or Tail, and when a dice is rolled, it can have six possible outcomes. So, there can be in total 2 x 6 = 12 possible events. Hence, the ample space can be written as:
S = { (H,1), (H,2), (H,3), (H,4), (H,5), (H,6),
(T,1), (T,2), (T,3), (T,4), (T,5), (T,6) }
问题8.投掷硬币，然后仅当硬币上显示头像时才旋转骰子。描述此实验的样本空间。
解决方案：
When a coin is tossed it can land on either of two sides Head or Tail, if the head turns up then we will roll the dice as given in the question. So, there will be in total 1 + 1 x 6 = 7 possible events. Hence, the sample space can be given as:
S = { T, (H,1), (H,2), (H,3), (H,4), (H,5), (H,6) }
问题9.硬币被扔了两次。如果第二次掷球导致尾巴，则掷出一个骰子。描述此实验的样本空间。
解决方案：
If a coin is tossed two times, then there can be four possible cases which are: HH, HT, TH, TT. Now, according to question, a dice is rolled only if second coin shows tails. So the total number of events associated with this event are:
2 x 6 + 2 = 14 events and sample space can be given as:
S = { (HT,1), (HT,2), (HT,3), (HT,4), (HT,5), (HT,6),
(TT,1), (TT,2), (TT,3), (TT,4), (TT,5), (TT,6), HH, TH }
问题10：一个实验包括抛硬币，如果发生头部抛掷，则第二次抛硬币。如果在第一次抛尾时出现尾巴，则一次掷一次骰子。找到样本空间。
解决方案：
When a coin is tossed and head comes then the coin is tossed again, so the sample space will be:
S1 = { HH, HT, TH, TT }
and when the coin is tossed and tail comes then the die is tossed, so the sample space will be:
S2 = { (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) }
Therefore, sample space for the entire experiment can be written as a union of these two sample space as:
S = S1 ∪ S2 = { HH, HT, TH, TT, (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) }
问题11：扔硬币。如果显示尾巴，则从包含2个红色3个黑色球的盒子中抽出一个球；如果显示头，我们就丢一个人。找到该实验的样本空间。
解决方案：
When a coin is tossed, we have two possible outcomes either head (H) or tail (T).
If head (H) turns up, we throw a die, then the sample space for this experiment is:
S1 = { (H,1), (H,2), (H,3), (H,4), (H,5), (H,6) }
and if tail (T) turns up, then we draw a ball from the box containing 2 red and 3 black balls, sample space for this experiment is:
S2 = { (T,R1), (T,R2), (T,B1), (T,B2), (T,B3) }
Therefore, sample space for the entire experiment can be written as a union of these two sample space as:
S = S1 ∪ S2 = { (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,R1), (T,R2), (T,B1), (T,B2), (T,B3) }
问题12.反复抛硬币直到第一次出现尾巴。编写此实验的样本空间。
解决方案：
According to the question, we will stop tossing the coin as soon as we get our first tail, else we will toss it repeatedly until we get a tail. Hence, the sample space for this experiment can be given as:
S = { T, HT, HHT, HHHT, ….. }