问题1.明天下雨的概率为0.85。明天不下雨的概率是多少?
解决方案:
Let E be the event of raining tomorrow
P(E) = 0.85 (given)
Sum of the probability of occurrence of an event and the probability of non-occurrence of an event is 1.
P(E) + P(not E) = 1
P(not E) = 1-0.85
Therefore, the probability that it will not rain tomorrow = 0.15
问题2.掷骰子。找到获得的可能性:
(i)质数
解决方案:
Total outcomes when a die is rolled are 1, 2, 3, 4, 5 and 6
Prime numbers are 2, 3, and 5
Probability = Number of favorable outcomes/ Total number of outcomes
Probability of getting a prime number = 3/6 = 1/2
(ii)2或4
解决方案:
Favorable outcomes = 2.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, Probability of getting 2 or 4 = 2/6 = 1/3
(iii)2或3的倍数
解决方案:
Multiple of 2 or 3 are 2, 3, 4 and 6.
Favorable outcomes = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, probability of getting multiple of 2 or 3 = 4/6 = 2/3
(iv)偶质数
解决方案:
Even prime number = 2
Favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting an even prime number = 1/6
(v)大于5的数字
解决方案:
Number greater than 5 = 6
Favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a number greater than 5 = 1/6
(vi)2到6之间的数字
解决方案:
Numbers lying between 2 and 6 are 3, 4 and 5
Favorable outcomes = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a number lying between 2 and 6 = 3/6 =1/2
问题3.将三枚硬币扔在一起。找到获得的可能性:
(i)正好两个脑袋
解决方案:
When three coins are tossed then the outcomes are TTT, THT, TTH, THH. HTT, HHT, HTH, HHH.
Total number of outcomes = 8.
Favorable outcome are THH, HHT, HTH
Number of favorable outcomes = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting exactly two heads = 3/8
(ii)至少两个首长
解决方案:
Favorable outcomes are HHT, HTH, HHH, THH
Number of favorable outcomes = 4.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting at least two heads = 4/8 = 1/2
(iii)至少一头一尾
解决方案:
Favorable outcomes are THT, TTH, THH, HTT, HHT, and HTH.
Number of favorable outcomes = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting at least one head and one tail = 6/8 = 3/4
(iv)没有尾巴
解决方案:
Favorable outcomes are HHH.
Number of favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting no tails is 1/8.
问题4. A和B扔一对骰子。如果A抛出9,则发现B抛出更高数字的机会
解决方案:
Total outcomes are (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),
Total outcomes = 36
Favorable outcomes are (5,5), (5,6), (6,4), (4,6), (6,5) and (6,6).
Number of favorable outcomes = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting the total of numbers on the dice greater than 9 = 6/36 = 1/6
问题5.掷出两个无偏的骰子。找出骰子上数字的总和大于10的概率。
解决方案:
Total outcomes are (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),
Total outcomes = 36
Favorable outcomes are (5, 6), (6, 5) and (6, 6).
Number of favorable events = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting the total of numbers on the dice greater than 10 = 3/36 = 1/12
问题6.从52张纸牌中随机抽取一张纸牌。找出开牌的概率为:
(i)黑人国王
解决方案:
Total number of cards = 52
Number of black king = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black king = 2/52 = 1/26
(ii)黑牌或国王
解决方案:
Total number of black cards are 26
Total number of kings which are not black = 2
Total number of black cards or king = 26+2 = 28
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black cards or a king = 28/52 = 7/13
(iii)黑与王
解决方案:
Total number of cards which are black and a king cards = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black cards and a king = 2/52 = 1/26
(iv)千斤顶,皇后或国王
解决方案:
A jack, queen or a king are 3 from each 4 suits.
Total number of a jack, queen and king = 12.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a jack, queen or a king is 12/52 = 3/13
(v)既不是一颗心也不是国王
解决方案:
Total number of cards that are a heart and a king = 13 + 3 = 16
Total number of cards that are neither a heart nor a king = 52 – 16 = 36
We know that, Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards neither a heart nor a king = 36/52 = 9/13
(vi)锹或王牌
解决方案:
Total number of spade cards = 13
Ace other than spade = 3
Total number of card which are spade or ace = 13 + 3 = 16
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards that is spade or an ace = 16/52 = 4/13
(vii)既不是王牌也不是国王
解决方案:
Total number of cards that are an ace or a king = 4 + 4 = 8
Total cards that are neither an ace nor a king = 52 – 8 = 44
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which are neither an ace nor a king = 44/52 = 11/13
(viii)既不是红牌也不是女王
解决方案:
Total red cards = 26.
Queens which are not red = 2
Total number of red cards or queen = 26 + 2 = 28
Total number of cards that are neither a red nor a queen= 52 -28 = 24
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting neither a red card nor a queen = 24/52 = 6/13
(ix)除ace外
解决方案:
Total number of ace = 4
Cards other than ace = 52-4 = 48
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of cards other than ace = 48/52 = 12/13
(x)十
解决方案:
Total number of tens = 4.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a ten = 4/52 = 1/13
(xi)一把铁锹
解决方案:
Total number of spade = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Thus, the probability of getting a spade = 13/52 = 1/4
(xii)黑卡
解决方案:
Total number of black cards = 26
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting black cards = 26/52 = 1/2
(xiii)七个俱乐部
解决方案:
Total number of 7 of club = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a 7 of club = 1/52
(xiv)插孔
解决方案:
Total number of jacks = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a jack = 4/52 = 1/13
(xv)黑桃王牌
解决方案:
Total number of ace of spade = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting an ace of spade = 1/52
(xvi)女王
解决方案:
Total number of queens = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a queen = 4/52 = 1/13
(xvii)一颗心
解决方案:
Total number of heart cards = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a heart card = 13/52 = 1/4
(xviii)红牌
解决方案:
Total number of red cards = 26
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red card = 26/52 = 1/2
问题7.在一张50张彩票中,编号从1到50的彩票中,一张彩票被抽中。找出抽取的票证带有质数的概率。
解决方案:
Total number of tickets = 50.
Tickets which are numbered as prime number are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Favorable outcomes = 15.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a prime number on the ticket = 15/50 = 3/10
问题8.骨灰盒包含10个红球和8个白球。随机抽出一个球。找出画出的球是白色的概率。
解决方案:
Total number of balls = 10 + 8 = 18
Total number of white balls = 8
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a white ball from the urn is 8/18 = 4/9
问题9.一个袋子包含3个红球,5个黑球和4个白球。从袋子中随机抽出一个球。抽出球的几率是:
(i)白色?
解决方案:
Total number of balls = 3 + 5 + 4 =12
Total white balls = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a white ball = 4/12 = 1/3
(ii)红色?
解决方案:
Total red balls = 3
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red ball = 3/12 = 1/4
(iii)黑色?
解决方案:
Total black balls is 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black ball = 5/12
(iv)不是红色
解决方案:
Number of balls which are not red are 4 white balls and 5 black balls = 4 + 5 = 9
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting no red ball = 9/12 = 3/4
问题10.从数字1、2、3,…,15中选择的数字是4的倍数的概率是多少?
解决方案:
Total numbers = 15
Numbers that are multiple of 4 are 4, 8 and 12.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of selecting a number which a multiple of 4 = 3/15 = 1/5
问题11.一个袋子包含6个红色,8个黑色和4个白色的球。随机抽出一个球。抽出的球不是黑色的概率是多少?
解决方案:
Total number of balls = 6 + 8 + 4 = 18
Total black balls = 8
Number of balls which are not black is = 18 – 8 = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a ball which is not black = 10/18 = 5/9
问题12.一个袋子包含5个白球和7个红球。随机抽出一个球。抽出的球是白色的几率是多少?
解决方案:
Total balls = 7 + 5 = 12
Total number of white balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a white ball = 5/12
问题13.从1到20的票证混合在一起,并随机抽取一张票证。开票的号码是3或7的倍数的机率是多少?
解决方案:
Total number of cards = 20.
Multiple of 3 or 7 are 3, 6, 7, 9, 12, 14, 15 and 18.
Favorable outcomes = 8.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a card that is a multiple of 3 or 7 = 8/20 = 2/5
问题14.在彩票中,有10个奖品和25个空白。获得奖品的几率是多少?
解决方案:
Total number of tickets = 10 + 25 = 35
Number of prize carrying tickets = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of winning a prize = 10/35 = 2/7
问题15.如果赢得一场比赛的机率是0.3,输掉该机率是多少?
解决方案:
Sum of probability of occurrence of an event and probability of non-occurrence of an event is 1.
Therefore, P(E) + P(not E) = 1
P(not E) = 1- 0.3
P(not E) = 0.7
Therefore, the probability of losing the game = 0.7
问题16.一个袋子包含5个黑色,7个红色和3个白色的球。从袋子中随机抽出一个球。找出被抽出的球的概率为:
(i)红色
解决方案:
Total balls = 7 + 5 + 3 = 15
Number red balls = 7
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a red ball = 7/15
(ii)黑色或白色
解决方案:
Total number of black or white balls = 5 + 3 = 8
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing white or black ball = 8/15
(iii)不是黑色
解决方案:
Total number of black balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing black ball P(E) = 5/15 = 1/3
Therefore, probability of not black = 1 – P(E)
= 1 – 1/3
= 2/3
问题17.一个袋子包含4个红色,5个黑色和6个白色的球。从袋子中随机抽出一个球。找出被抽出的球的概率为:
(i)白色
解决方案:
Total number of balls = 4 + 5 + 6 = 15
Total white balls = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing white a ball = 6/15 = 2/5
(ii)红色
解决方案:
Total number of red balls = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing red a ball = 4/15
(iii)不是黑色
解决方案:
Total number of black balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing black ball P(E) = 5/15 = 1/3
Therefore, probability of not black = 1 – P(E)
= 1 – 1/3
= 2/3
Therefore, the probability of drawing a ball that is not black is 2/3
(iv)红色或白色
解决方案:
Total number of red or white balls= 4 + 6 = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a white or red ball = 10/15 = 2/3
问题18:一张纸牌是从一张52张纸牌中抽调得很好的。找到获得的可能性:
(i)红色西装的国王
解决方案:
Total number of cards = 52
Total cards which are king of red suit = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which is a king of red suit = 2/52 = 1/26
(ii)面卡
解决方案:
Total number of face cards = 12
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a face card = 12/52 = 3/13
(iii)红脸卡
解决方案:
Total number of red face cards = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red face card = 6/52 = 3/26
(iv)黑衣皇后
解决方案:
Total number of queen of black suit cards = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which is a queen of black suit = 2/52 = 1/26
(v)千真万确
解决方案:
Total number of jack of hearts = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a card which is a jack of hearts = 1/52
(vi)一把铁锹
解决方案:
Total number of spade cards = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a spade card = 13/52 = 1/4