从一副 52 张牌中抽到红心或 J 的概率是多少？
处理随机事件发生的数学分支称为概率。它在数学中用于预测事件发生的可能性。任何事件的概率只能在 0 到 1 之间，也可以写成百分比的形式。
可能性
事件 A 的概率一般写为 P(A)。这里，P 代表可能性，A 代表事件。它说明事件即将发生的可能性。事件的概率只能存在于 0 和 1 之间，其中 0 表示事件不会发生，即不可能，1 表示肯定会发生，即确定性。
如果不确定事件的结果，请借助某些结果的概率，以及它们发生的可能性。为了正确理解概率，以抛硬币为例，会有两种可能的结果——正面或反面。
概率公式
Probability of an event, P(A) = Favorable outcomes / Total number of outcomes
概率论的一些术语
概率中有很多不常用的术语，如实验、样本空间、有利结果、试验、随机实验等。让我们详细看一下它们的定义，
- 实验：为产生结果而进行的操作或试验称为实验。
- 样本空间：一个实验共同构成了所有可能结果的样本空间。例如，抛硬币的样本空间是正面和反面。
- 有利结果：产生所需结果的事件称为有利结果。例如，如果同时掷出两个骰子，则将两个骰子上的数字之和设为 4 的可能或有利结果是 (1, 3)、(2, 2) 和 (3, 1)。
- 试验：试验意味着进行随机实验。
- 随机实验：随机实验是具有明确定义的结果集的实验。例如，当抛硬币时，会得到正面或反面，但结果不确定哪个会出现。
- 事件：事件是随机实验的结果。
- 同等可能的事件：同等可能的事件是具有相同机会或概率发生的罕见事件。这里一个事件的结果独立于另一个。例如，当抛硬币时，得到正面或反面的机会均等。
- 穷举事件：穷举事件是当实验的所有结果的集合等于样本空间时。
- 互斥事件：不能同时发生的事件称为互斥事件。例如，气候可以是冷的或热的。一个人不能一次又一次地经历同样的天气。
The Possibility of only two outcomes which is an event will occur or not, like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.
一些概率公式
- 加法规则：两个事件的并集，比如 A 和 B，那么，
P(A or B) = P(A) + P(B) – P(A∩B)
P(A∪ B) = P(A) + P(B) – P(A∩B)
- 互补规则：如果一个实验有两个可能的事件，那么一个事件的概率将是另一个事件的补码。例如，如果 A 和 B 是两个可能的事件，那么，
P(B) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
- 条件规则：当给定一个事件的概率并且需要第二个事件的概率时，第一个给定，那么 P(B, given A) = P(A and B), P(A, given B)。反之亦然，
P(B∣A) = P(A∩B)/P(A)
- 乘法规则：另外两个事件的交集，即事件 A 和 B 需要同时发生。然后
P(A and B) = P(A)⋅P(B).
P(A∩B) = P(A)⋅P(B∣A)
从一副 52 张牌中抽到红心或 J 的概率是多少？
解决方案：
It is known that a well-shuffled deck has 52 cards
Total number of black cards = 26
Total number of red cards = 26
further divided into suits (4 of them: Spades, Hearts, Diamonds, Clubs) of 13 cards each.
And Each suit has 13 cards (A, 2 to10, Jack, Queen, King).
So , total number of outcome = 52
probability of getting either a heart or a jack?
probability of getting a heart = 13
probability of getting a jack = 4
And probability of getting a jack of heart = 1
Therefore probability of getting a heart = {total number of heart cards in the deck}/{total number of cards in the deck}
= 13/52
Probability of getting a heart = 1/4
And the probability of getting either a jack = {total number of jack cards in the deck}/{total number of cards in the deck}
= 4/52
Probability of getting a jack = 1/13
probability of getting a jack of heart = {total number of jack of heart in the deck}/{total number of cards in the deck}
= 1/52
类似问题
问题1：得到皇后或红牌的概率是多少？
解决方案：
Total number of cards are 52
number of red cards are 26 and queens are 4 whereas 26 red cards contain 2 queens(so only 2 will be considered out of 4).
So, total outcomes = 52
favorable outcomes = 26 + 2 = 28
So, the probability of getting a queen or red card = Favorable outcomes / Total outcomes
= 28 / 52 = 7/13
P = 7/13
问题 2：从一副洗好的 52 张牌中抽到一张黑牌的概率是多少？
解决方案：
We know that a well-shuffled deck has 52 cards
Total number of black cards = 26
Total number of red cards = 26
Therefore probability of getting a black card= {total number of black cards in the deck}/{total number of cards in the deck}
= 26/52
= 1/2
So the probability of having black card is 1/2
问题3：得到黑皇后或钻石的概率是多少？
解决方案：
Total number of cards = 52
Number of favorable cards that are black queen = 2
so, probability of getting a black queen = 2/52
Total number of cards that are diamond =13
Therefore probability of getting a diamond = {13/52}
Therefore, probability of getting a black queen = 2/52
P(E) = probability of getting a black queen + probability of getting a diamond
= 2/52 +13/52
= 15/32
问题 4：求一次掷骰子得到小于 5 的数字的概率。
解决方案：
When the dice is rolled then there will be 6 outcomes.
Total number of favorable outcome {set of outcome} = {1, 2, 3, 4, 5, 6}
= 6
Now as per the question,
Probability of getting a number less than 5 in a single throw is 4
Numbers less than 5 are {1,2,3,4}
therefore favorable outcome will be = 4
P(A) = Favorable outcomes / Total number of outcomes
= 4/6
= 2/3
Hence the probability of getting a number less than 5 in a single throw of a die is 2/3
问题5：连续翻转7个正面的几率是多少？
解决方案：
Probability of an event = (number of favorable event) / (total number of event).
P(B) = (occurrence of Event B) / (total number of event).
Probability of getting one head = 1/2.
here Tossing a coin is an independent event, its not dependent on how many times it has been tossed.
Probability of getting 2 heads in a row = probability of getting head first time × probability of getting head second time.
Probability of getting 2 head in a row = (1/2) × (1/2)
Therefore, the probability of flipping 7 heads in a row = (1/2)7