E.of.B <- function( F, G ) { return( 100 * ( F + G ) * ( 1 - F / ( 4 * ( F + G ) ) )^ ( ceiling( ( F + G ) / 20 ) ) ) } plot( 0:100, E.of.B( 0:100, 25 ), type = 'l', lwd = 2, col = 'red', xlab = 'Number of fake notes', ylab = 'Expected value of bank account' ) ( 0:100 )[ E.of.B( 0:100, 25 ) == max( E.of.B( 0:100, 25 ) ) ] # [1] 55 E.of.B( 55, 25 ) - 2500 # [1] 1262.475 [,1] [,2] [1,] 0 2500.000 [2,] 1 2550.240 [3,] 2 2600.926 [4,] 3 2652.009 [5,] 4 2703.448 [6,] 5 2755.208 [7,] 6 2807.258 [8,] 7 2859.570 [9,] 8 2912.121 [10,] 9 2964.890 [11,] 10 3017.857 [12,] 11 3071.007 [13,] 12 3124.324 [14,] 13 3177.796 [15,] 14 3231.410 [16,] 15 3285.156 [17,] 16 3013.266 [18,] 17 3049.666 [19,] 18 3086.351 [20,] 19 3123.299 [21,] 20 3160.494 [22,] 21 3197.917 [23,] 22 3235.553 [24,] 23 3273.389 [25,] 24 3311.412 [26,] 25 3349.609 [27,] 26 3387.971 [28,] 27 3426.487 [29,] 28 3465.148 [30,] 29 3503.945 [31,] 30 3542.872 [32,] 31 3581.920 [33,] 32 3621.083 [34,] 33 3660.356 [35,] 34 3699.731 [36,] 35 3739.204 [37,] 36 3221.246 [38,] 37 3248.739 [39,] 38 3276.374 [40,] 39 3304.145 [41,] 40 3332.044 [42,] 41 3360.065 [43,] 42 3388.203 [44,] 43 3416.451 [45,] 44 3444.805 [46,] 45 3473.261 [47,] 46 3501.812 [48,] 47 3530.455 [49,] 48 3559.187 [50,] 49 3588.002 [51,] 50 3616.898 [52,] 51 3645.871 [53,] 52 3674.919 [54,] 53 3704.037 [55,] 54 3733.223 [56,] 55 3762.475 <- [57,] 56 3136.418 [58,] 57 3157.120 [59,] 58 3177.902 [60,] 59 3198.761 [61,] 60 3219.693 [62,] 61 3240.697 [63,] 62 3261.769 [64,] 63 3282.907 [65,] 64 3304.108 [66,] 65 3325.371 [67,] 66 3346.692 [68,] 67 3368.071 [69,] 68 3389.504 [70,] 69 3410.990 [71,] 70 3432.528 [72,] 71 3454.115 [73,] 72 3475.750 [74,] 73 3497.431 [75,] 74 3519.157 [76,] 75 3540.926 [77,] 76 2892.519 [78,] 77 2908.085 [79,] 78 2923.699 [80,] 79 2939.360 [81,] 80 2955.065 [82,] 81 2970.814 [83,] 82 2986.605 [84,] 83 3002.437 [85,] 84 3018.308 [86,] 85 3034.218 [87,] 86 3050.165 [88,] 87 3066.148 [89,] 88 3082.167 [90,] 89 3098.219 [91,] 90 3114.305 [92,] 91 3130.424 [93,] 92 3146.573 [94,] 93 3162.753 [95,] 94 3178.963 [96,] 95 3195.201 [97,] 96 2574.482 [98,] 97 2586.178 [99,] 98 2597.903 [100,] 99 2609.657 [101,] 100 2621.440