Príklad o narodeninách a lístku do kina

n <- 10
n*prod(seq(from = 364, by = -1, length.out = n-1))/(365)^n

## [1] 0.0241932

# ?polyroot
polyroot(c(365, 363, -1))

## [1]  -1.00274+0i 364.00274+0i

n <- 365
n*prod(seq(from = 364, by = -1, length.out = n-1))/(365)^n # numericky problem

## [1] NaN

pn <- function(n) n*prod(seq(from = 364, by = -1, length.out = n-1))/(365)^n
pn(10)

## [1] 0.0241932

pn(15)

## [1] 0.03070269

pn(20)

## [1] 0.03224995

pn(30)

## [1] 0.02413839

polyroot(c(365, -1, -1)) # opravena kvadraticka rovnica

## [1]  18.61151+0i -19.61151+0i

pn(18)

## [1] 0.03220711

pn(19)

## [1] 0.03231986

pn(20)

## [1] 0.03224995

pn(300)

## [1] NaN

n <- 10
n*prod(seq(from = 364, by = -1, length.out = n-1))/(365)^n

## [1] 0.0241932

seq(from = 364, by = -1, length.out = n-1)

## [1] 364 363 362 361 360 359 358 357 356

log(seq(from = 364, by = -1, length.out = n-1))

## [1] 5.897154 5.894403 5.891644 5.888878 5.886104 5.883322 5.880533 5.877736
## [9] 5.874931

sum(log(seq(from = 364, by = -1, length.out = n-1)))

## [1] 52.9747

pn2 <- function(n) {
  log_pn <- log(n) + sum(log(seq(from = 364, by = -1, length.out = n-1))) - n*log(365)
  return(exp(log_pn))
}
pn2(10)

## [1] 0.0241932

pn(10)

## [1] 0.0241932

pn2(300)

## [1] 5.13315e-82

Ostatné príklady

19/36

## [1] 0.5277778

(3/4)^(1:10)

##  [1] 0.75000000 0.56250000 0.42187500 0.31640625 0.23730469 0.17797852
##  [7] 0.13348389 0.10011292 0.07508469 0.05631351

(0.15*0.75)/(0.15*0.75 + 0.25*0.85) # 0.3461538

## [1] 0.3461538