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!(20−5)!=20⋅19⋅18⋅17⋅165⋅4⋅3⋅2⋅1=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=7.10637079329⋅1040 b1=200⋅ 199⋅ 198⋅ 197⋅ 196⋅ 195⋅ 194⋅ 193⋅ 192⋅ 191⋅ 190⋅ 189⋅ 188⋅ 187⋅ 186⋅ 185⋅ 184⋅ 183⋅ 182⋅ 181=3.92570096972⋅1045 a2=20⋅ 19⋅ 18⋅ 17⋅ 16⋅ 180⋅ 179⋅ 178⋅ 177⋅ 176⋅ 175⋅ 174⋅ 173⋅ 172⋅ 171⋅ 170⋅ 169⋅ 168⋅ 167⋅ 166=6.8910262238⋅1039  p1=100⋅ n⋅ a1b1=100⋅ 15504⋅ 7.10637079329⋅10403.92570096972⋅1045≐28.0656 %  p2=100⋅ n⋅ a2b1=100⋅ 15504⋅ 6.8910262238⋅10393.92570096972⋅1045≐2.7215 % p=p1+p2=28.0656+2.7215≐30.7871≐30.78712%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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