＊DSE Maths/AL M&S: Probability 概率

題目: In a city, there is 45% people who support People Power. Others support LYL. In those people who support People Power, 70% of them are clever, others are stupid. In those people who support LYL, 80% of them are stupid, others are clever.
有45%人支持人民力量,其他支持禮義廉;人民力量支持者入面有70%既人係聰明,其他係蠢;禮義廉支持者入面有80%係蠢,其他係聰明.

a) what is the probability that a chosen person is stupid?and clever?

好簡單~畫個tree diagram出黎就一了百了~
ans: stupid: 0.45*0.3 + 0.55*0.8 = 0.575; clever: 1-0.575 = 0.425(因為唔係stupid就係clever)

b) if a chosen person is LYL fans, what is the probability that he is stupid?

ans: 0.55*0.8 / 0.55, 其實即係0.8~
見到"if xxxxx", "suppose xxxx," 就一定係conditional probability喇~題目if"乜野",就擺"乜野"做分母~

c) if a chosen person is stupid, what is the probability that he is LYL fans?

ans:( 0.55*0.8 ) / (0.55*0.8 + 0.45*0.3) = 0.7652
分母係stupid, 分子係stupid & LYL fans(分子唔好漏左stupid個part哦),
可以見到..conditional probability好多時個分子可以係個分母度搵出黎(0.55*0.8)~

d) if 5 people are chosen, what is probability that there is exactly 4 LYL fans?

ans: 5C1 * 0.55^4 * 0.45 = 0.2059

好普通既Binomial..但上面講左只要係if xxxx就一定係conditional probability, 咁呢條點解唔洗除?

因為係if 5 people are chosen, 由頭到尾都冇乘過5 people are chosen既prob, 又洗乜除呢~

e) if 5 stupid people are chosen, what is probability that there is at least 3 LYL fans?

方法一: 分母(conditional prob習慣諗左分母先..): (0.575)^5
分子: at least 3 LYL & stupid, others are People Power & stupid:

5C3 * (0.55*0.8)^3 (0.45*0.3)^2 + 5C4 * (0.55*0.8)^4 (0.45*0.3)^1 + 

5C5 * (0.55*0.8)^5 (0.45*0.3)^0

分子除分母 = 0.9119

方法二: 5C3 * 0.7652^3 * (1-0.7652)^2 + 5C4 * 0.7652^4 * (1-0.7652) + 0.7652^5 = 0.9119
呢個方法唔洗除, 因為given左5個都係stupid..直接用(c) part個conditional prob(0.7652)做呢個Binomial既p就ok鳥~

如果given左個condition只係得一個情況(5個stupid),可以試下用唔用到唔洗除既方法

f) if there is at least 3 stupid people out of 5 chosen people, what is the probability that exactly 3 of those stupid people is LYL fans?

如果個condition係at least呢..即係有幾個情況(3個stupid,4個stupid,5個stupid),咁就一定要用除既方法..(e part既方法一)

ans:
分母: 5C3 * 0.575^3 * 0.425^2 + 5C4 * 0.575^4 * 0.425^1 + 5C5 * 0.575^5 * 0.425^0 = 0.6385
分子:
3個LYLstupid & 2個clever(我冇限clever既係LYL定PP): 5C3 * (0.55*0.8)^3 * 0.425^2 +
3個LYLstupid & 1個係PP stupid & 1個係clever: 5C3 * (0.55*0.8)^3 * 2C1 * (0.45*0.3) * 0.425 +
3個LYLstupid & 2個係PP stupid: 5C3 * (0.55*0.8)^3 * (0.45*0.3)^2
分子除分母 = 0.4184 (希望冇計錯啦sosad)