Sick days

In Canada, there are typically 261 working days per year. If there is a 4.9% chance that an employee takes a sick day. ..
what is the probability an employee will use 17 OR MORE sick days in a year?

Result

p =  0.093

Solution:

q=4.9%=4.9100=0.049 n=261  C0(261)=(2610)=261!0!(261−0)!=11=1  p0=(2610)⋅ q0⋅ (1−q)n−0=1⋅ 0.0490⋅ (1−0.049)261−0≐0 C1(261)=(2611)=261!1!(261−1)!=2611=261  p1=(2611)⋅ q1⋅ (1−q)n−1=261⋅ 0.0491⋅ (1−0.049)261−1≐0 C2(261)=(2612)=261!2!(261−2)!=261⋅2602⋅1=33930  p2=(2612)⋅ q2⋅ (1−q)n−2=33930⋅ 0.0492⋅ (1−0.049)261−2≐0.0002 C3(261)=(2613)=261!3!(261−3)!=261⋅260⋅2593⋅2⋅1=2929290  p3=(2613)⋅ q3⋅ (1−q)n−3=2929290⋅ 0.0493⋅ (1−0.049)261−3≐0.0008 C4(261)=(2614)=261!4!(261−4)!=261⋅260⋅259⋅2584⋅3⋅2⋅1=188939205  p4=(2614)⋅ q4⋅ (1−q)n−4=188939205⋅ 0.0494⋅ (1−0.049)261−4≐0.0027 C5(261)=(2615)=261!5!(261−5)!=261⋅260⋅259⋅258⋅2575⋅4⋅3⋅2⋅1=9711475137  p5=(2615)⋅ q5⋅ (1−q)n−5=9711475137⋅ 0.0495⋅ (1−0.049)261−5≐0.0071 C6(261)=(2616)=261!6!(261−6)!=261⋅260⋅259⋅258⋅257⋅2566⋅5⋅4⋅3⋅2⋅1=414356272512  p6=(2616)⋅ q6⋅ (1−q)n−6=414356272512⋅ 0.0496⋅ (1−0.049)261−6≐0.0157 C7(261)=(2617)=261!7!(261−7)!=261⋅260⋅259⋅258⋅257⋅256⋅2557⋅6⋅5⋅4⋅3⋅2⋅1=15094407070080  p7=(2617)⋅ q7⋅ (1−q)n−7=15094407070080⋅ 0.0497⋅ (1−0.049)261−7≐0.0294 C8(261)=(2618)=261!8!(261−8)!=261⋅260⋅259⋅258⋅257⋅256⋅255⋅2548⋅7⋅6⋅5⋅4⋅3⋅2⋅1=479247424475040  p8=(2618)⋅ q8⋅ (1−q)n−8=479247424475040⋅ 0.0498⋅ (1−0.049)261−8≐0.0481 C9(261)=(2619)=261!9!(261−9)!=261⋅260⋅259⋅258⋅257⋅256⋅255⋅254⋅2539⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1=13472177599131680  p9=(2619)⋅ q9⋅ (1−q)n−9=13472177599131680⋅ 0.0499⋅ (1−0.049)261−9≐0.0696 C10(261)=(26110)=261!10!(261−10)!=339498875498118336  p10=(26110)⋅ q10⋅ (1−q)n−10=339498875498118336⋅ 0.04910⋅ (1−0.049)261−10≐0.0904 C11(261)=(26111)=261!11!(261−11)!=7746747068184336576  p11=(26111)⋅ q11⋅ (1−q)n−11=7746747068184336576⋅ 0.04911⋅ (1−0.049)261−11≐0.1063 C12(261)=(26112)=261!12!(261−12)!≈1.613×1020=161390563920507012000  p12=(26112)⋅ q12⋅ (1−q)n−12=161390563920507012000⋅ 0.04912⋅ (1−0.049)261−12≐0.1141 C13(261)=(26113)=261!13!(261−13)!≈3.091×1021=3091250032015865076000  p13=(26113)⋅ q13⋅ (1−q)n−13=3091250032015865076000⋅ 0.04913⋅ (1−0.049)261−13≐0.1126 C14(261)=(26114)=261!14!(261−14)!≈5.475×1022=54759286281423895632000  p14=(26114)⋅ q14⋅ (1−q)n−14=54759286281423895632000⋅ 0.04914⋅ (1−0.049)261−14≐0.1028 C15(261)=(26115)=261!15!(261−15)!≈9.017×1023=901702914100780148073600  p15=(26115)⋅ q15⋅ (1−q)n−15=901702914100780148073600⋅ 0.04915⋅ (1−0.049)261−15≐0.0872 C16(261)=(26116)=261!16!(261−16)!≈1.386×1025=13863682304299494776631600  p16=(26116)⋅ q16⋅ (1−q)n−16=13863682304299494776631600⋅ 0.04916⋅ (1−0.049)261−16≐0.0691 C17(261)=(26117)=261!17!(261−17)!≈1.998×1026=199800127326669189427926000  p17=(26117)⋅ q17⋅ (1−q)n−17=199800127326669189427926000⋅ 0.04917⋅ (1−0.049)261−17≐0.0513  s=p0+p1+p2+p3+p4+p5+p6+p7+p8+p9+p10+p11+p12+p13+p14+p15+p16+p17=0+0+0.0002+0.0008+0.0027+0.0071+0.0157+0.0294+0.0481+0.0696+0.0904+0.1063+0.1141+0.1126+0.1028+0.0872+0.0691+0.0513≐0.9071 p=1−s=1−0.9071≐0.0929≐0.093q=4.9 \%=\dfrac{ 4.9 }{ 100 }=0.049 \ \\ n=261 \ \\ \ \\ C_{{ 0}}(261)=\dbinom{ 261}{ 0}=\dfrac{ 261! }{ 0!(261-0)!}=\dfrac{ 1 } { 1 }=1 \ \\ \ \\ p_{0}={ { 261 } \choose 0 } \cdot \ q^{ 0 } \cdot \ (1-q)^{ n-0 }=1 \cdot \ 0.049^{ 0 } \cdot \ (1-0.049)^{ 261-0 } \doteq 0 \ \\ C_{{ 1}}(261)=\dbinom{ 261}{ 1}=\dfrac{ 261! }{ 1!(261-1)!}=\dfrac{ 261 } { 1 }=261 \ \\ \ \\ p_{1}={ { 261 } \choose 1 } \cdot \ q^{ 1 } \cdot \ (1-q)^{ n-1 }=261 \cdot \ 0.049^{ 1 } \cdot \ (1-0.049)^{ 261-1 } \doteq 0 \ \\ C_{{ 2}}(261)=\dbinom{ 261}{ 2}=\dfrac{ 261! }{ 2!(261-2)!}=\dfrac{ 261 \cdot 260 } { 2 \cdot 1 }=33930 \ \\ \ \\ p_{2}={ { 261 } \choose 2 } \cdot \ q^{ 2 } \cdot \ (1-q)^{ n-2 }=33930 \cdot \ 0.049^{ 2 } \cdot \ (1-0.049)^{ 261-2 } \doteq 0.0002 \ \\ C_{{ 3}}(261)=\dbinom{ 261}{ 3}=\dfrac{ 261! }{ 3!(261-3)!}=\dfrac{ 261 \cdot 260 \cdot 259 } { 3 \cdot 2 \cdot 1 }=2929290 \ \\ \ \\ p_{3}={ { 261 } \choose 3 } \cdot \ q^{ 3 } \cdot \ (1-q)^{ n-3 }=2929290 \cdot \ 0.049^{ 3 } \cdot \ (1-0.049)^{ 261-3 } \doteq 0.0008 \ \\ C_{{ 4}}(261)=\dbinom{ 261}{ 4}=\dfrac{ 261! }{ 4!(261-4)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 } { 4 \cdot 3 \cdot 2 \cdot 1 }=188939205 \ \\ \ \\ p_{4}={ { 261 } \choose 4 } \cdot \ q^{ 4 } \cdot \ (1-q)^{ n-4 }=188939205 \cdot \ 0.049^{ 4 } \cdot \ (1-0.049)^{ 261-4 } \doteq 0.0027 \ \\ C_{{ 5}}(261)=\dbinom{ 261}{ 5}=\dfrac{ 261! }{ 5!(261-5)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 \cdot 257 } { 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=9711475137 \ \\ \ \\ p_{5}={ { 261 } \choose 5 } \cdot \ q^{ 5 } \cdot \ (1-q)^{ n-5 }=9711475137 \cdot \ 0.049^{ 5 } \cdot \ (1-0.049)^{ 261-5 } \doteq 0.0071 \ \\ C_{{ 6}}(261)=\dbinom{ 261}{ 6}=\dfrac{ 261! }{ 6!(261-6)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 \cdot 257 \cdot 256 } { 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=414356272512 \ \\ \ \\ p_{6}={ { 261 } \choose 6 } \cdot \ q^{ 6 } \cdot \ (1-q)^{ n-6 }=414356272512 \cdot \ 0.049^{ 6 } \cdot \ (1-0.049)^{ 261-6 } \doteq 0.0157 \ \\ C_{{ 7}}(261)=\dbinom{ 261}{ 7}=\dfrac{ 261! }{ 7!(261-7)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 \cdot 257 \cdot 256 \cdot 255 } { 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=15094407070080 \ \\ \ \\ p_{7}={ { 261 } \choose 7 } \cdot \ q^{ 7 } \cdot \ (1-q)^{ n-7 }=15094407070080 \cdot \ 0.049^{ 7 } \cdot \ (1-0.049)^{ 261-7 } \doteq 0.0294 \ \\ C_{{ 8}}(261)=\dbinom{ 261}{ 8}=\dfrac{ 261! }{ 8!(261-8)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 \cdot 257 \cdot 256 \cdot 255 \cdot 254 } { 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=479247424475040 \ \\ \ \\ p_{8}={ { 261 } \choose 8 } \cdot \ q^{ 8 } \cdot \ (1-q)^{ n-8 }=479247424475040 \cdot \ 0.049^{ 8 } \cdot \ (1-0.049)^{ 261-8 } \doteq 0.0481 \ \\ C_{{ 9}}(261)=\dbinom{ 261}{ 9}=\dfrac{ 261! }{ 9!(261-9)!}=\dfrac{ 261 \cdot 260 \cdot 259 \cdot 258 \cdot 257 \cdot 256 \cdot 255 \cdot 254 \cdot 253 } { 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=13472177599131680 \ \\ \ \\ p_{9}={ { 261 } \choose 9 } \cdot \ q^{ 9 } \cdot \ (1-q)^{ n-9 }=13472177599131680 \cdot \ 0.049^{ 9 } \cdot \ (1-0.049)^{ 261-9 } \doteq 0.0696 \ \\ C_{{ 10}}(261)=\dbinom{ 261}{ 10}=\dfrac{ 261! }{ 10!(261-10)!}=339498875498118336 \ \\ \ \\ p_{10}={ { 261 } \choose 10 } \cdot \ q^{ 10 } \cdot \ (1-q)^{ n-10 }=339498875498118336 \cdot \ 0.049^{ 10 } \cdot \ (1-0.049)^{ 261-10 } \doteq 0.0904 \ \\ C_{{ 11}}(261)=\dbinom{ 261}{ 11}=\dfrac{ 261! }{ 11!(261-11)!}=7746747068184336576 \ \\ \ \\ p_{11}={ { 261 } \choose 11 } \cdot \ q^{ 11 } \cdot \ (1-q)^{ n-11 }=7746747068184336576 \cdot \ 0.049^{ 11 } \cdot \ (1-0.049)^{ 261-11 } \doteq 0.1063 \ \\ C_{{ 12}}(261)=\dbinom{ 261}{ 12}=\dfrac{ 261! }{ 12!(261-12)!} \approx 1.613\times 10^{ 20 }=161390563920507012000 \ \\ \ \\ p_{12}={ { 261 } \choose 12 } \cdot \ q^{ 12 } \cdot \ (1-q)^{ n-12 }=161390563920507012000 \cdot \ 0.049^{ 12 } \cdot \ (1-0.049)^{ 261-12 } \doteq 0.1141 \ \\ C_{{ 13}}(261)=\dbinom{ 261}{ 13}=\dfrac{ 261! }{ 13!(261-13)!} \approx 3.091\times 10^{ 21 }=3091250032015865076000 \ \\ \ \\ p_{13}={ { 261 } \choose 13 } \cdot \ q^{ 13 } \cdot \ (1-q)^{ n-13 }=3091250032015865076000 \cdot \ 0.049^{ 13 } \cdot \ (1-0.049)^{ 261-13 } \doteq 0.1126 \ \\ C_{{ 14}}(261)=\dbinom{ 261}{ 14}=\dfrac{ 261! }{ 14!(261-14)!} \approx 5.475\times 10^{ 22 }=54759286281423895632000 \ \\ \ \\ p_{14}={ { 261 } \choose 14 } \cdot \ q^{ 14 } \cdot \ (1-q)^{ n-14 }=54759286281423895632000 \cdot \ 0.049^{ 14 } \cdot \ (1-0.049)^{ 261-14 } \doteq 0.1028 \ \\ C_{{ 15}}(261)=\dbinom{ 261}{ 15}=\dfrac{ 261! }{ 15!(261-15)!} \approx 9.017\times 10^{ 23 }=901702914100780148073600 \ \\ \ \\ p_{15}={ { 261 } \choose 15 } \cdot \ q^{ 15 } \cdot \ (1-q)^{ n-15 }=901702914100780148073600 \cdot \ 0.049^{ 15 } \cdot \ (1-0.049)^{ 261-15 } \doteq 0.0872 \ \\ C_{{ 16}}(261)=\dbinom{ 261}{ 16}=\dfrac{ 261! }{ 16!(261-16)!} \approx 1.386\times 10^{ 25 }=13863682304299494776631600 \ \\ \ \\ p_{16}={ { 261 } \choose 16 } \cdot \ q^{ 16 } \cdot \ (1-q)^{ n-16 }=13863682304299494776631600 \cdot \ 0.049^{ 16 } \cdot \ (1-0.049)^{ 261-16 } \doteq 0.0691 \ \\ C_{{ 17}}(261)=\dbinom{ 261}{ 17}=\dfrac{ 261! }{ 17!(261-17)!} \approx 1.998\times 10^{ 26 }=199800127326669189427926000 \ \\ \ \\ p_{17}={ { 261 } \choose 17 } \cdot \ q^{ 17 } \cdot \ (1-q)^{ n-17 }=199800127326669189427926000 \cdot \ 0.049^{ 17 } \cdot \ (1-0.049)^{ 261-17 } \doteq 0.0513 \ \\ \ \\ s=p_{0}+ p_{1}+ p_{2}+ p_{3}+ p_{4}+ p_{5}+ p_{6}+ p_{7}+ p_{8}+ p_{9}+ p_{10}+ p_{11}+ p_{12}+ p_{13}+ p_{14}+ p_{15}+ p_{16}+ p_{17}=0+ 0+ 0.0002+ 0.0008+ 0.0027+ 0.0071+ 0.0157+ 0.0294+ 0.0481+ 0.0696+ 0.0904+ 0.1063+ 0.1141+ 0.1126+ 0.1028+ 0.0872+ 0.0691+ 0.0513 \doteq 0.9071 \ \\ p=1-s=1-0.9071 \doteq 0.0929 \doteq 0.093

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