Raffle

There are 200 draws in the raffle, but only 20 of them win. What is the probability of at least 4 winnings for a group of people who have bought 5 tickets together?

Result

p =  30.78712 %

Solution:

C5(20)=(205)=20!5!(205)!=201918171654321=15504  n=(205)=15504 a1=20 19 18 17 180 179 178 177 176 175 174 173 172 171 170 169 168 167 166 165=7.106370793291040 b1=200 199 198 197 196 195 194 193 192 191 190 189 188 187 186 185 184 183 182 181=3.925700969721045 a2=20 19 18 17 16 180 179 178 177 176 175 174 173 172 171 170 169 168 167 166=6.89102622381039  p1=100 n a1b1=100 15504 7.1063707932910403.92570096972104528.0656 %  p2=100 n a2b1=100 15504 6.891026223810393.9257009697210452.7215 % p=p1+p2=28.0656+2.721530.787130.78712%C_{{ 5}}(20)=\dbinom{ 20}{ 5}=\dfrac{ 20! }{ 5!(20-5)!}=\dfrac{ 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 } { 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=15504 \ \\ \ \\ n={ { 20 } \choose 5 }=15504 \ \\ a_{1}=20 \cdot \ 19 \cdot \ 18 \cdot \ 17 \cdot \ 180 \cdot \ 179 \cdot \ 178 \cdot \ 177 \cdot \ 176 \cdot \ 175 \cdot \ 174 \cdot \ 173 \cdot \ 172 \cdot \ 171 \cdot \ 170 \cdot \ 169 \cdot \ 168 \cdot \ 167 \cdot \ 166 \cdot \ 165=7.10637079329\cdot 10^{ 40 } \ \\ b_{1}=200 \cdot \ 199 \cdot \ 198 \cdot \ 197 \cdot \ 196 \cdot \ 195 \cdot \ 194 \cdot \ 193 \cdot \ 192 \cdot \ 191 \cdot \ 190 \cdot \ 189 \cdot \ 188 \cdot \ 187 \cdot \ 186 \cdot \ 185 \cdot \ 184 \cdot \ 183 \cdot \ 182 \cdot \ 181=3.92570096972\cdot 10^{ 45 } \ \\ a_{2}=20 \cdot \ 19 \cdot \ 18 \cdot \ 17 \cdot \ 16 \cdot \ 180 \cdot \ 179 \cdot \ 178 \cdot \ 177 \cdot \ 176 \cdot \ 175 \cdot \ 174 \cdot \ 173 \cdot \ 172 \cdot \ 171 \cdot \ 170 \cdot \ 169 \cdot \ 168 \cdot \ 167 \cdot \ 166=6.8910262238\cdot 10^{ 39 } \ \\ \ \\ p_{1}=100 \cdot \ n \cdot \ \dfrac{ a_{1} }{ b_{1} }=100 \cdot \ 15504 \cdot \ \dfrac{ 7.10637079329 \cdot 10^{40} }{ 3.92570096972 \cdot 10^{45} } \doteq 28.0656 \ \% \ \\ \ \\ p_{2}=100 \cdot \ n \cdot \ \dfrac{ a_{2} }{ b_{1} }=100 \cdot \ 15504 \cdot \ \dfrac{ 6.8910262238 \cdot 10^{39} }{ 3.92570096972 \cdot 10^{45} } \doteq 2.7215 \ \% \ \\ p=p_{1}+p_{2}=28.0656+2.7215 \doteq 30.7871 \doteq 30.78712 \%



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