Rapanga: Te Tepe Tēpu

He Pārekareka te Pāngarau

Me whakamātau koe i ēnei i te tuatahi. Kua kōpakina ia otinga; whakatuwherahia ina rite koe.

1. Ka tīmata tētahi MartingalerautakiMartingaleHe kauneke e whakarua ana i te peti i muri i ia hinga. Tirohia te wāhanga Rautaki mō ōna mahi me ngā mea kāore e mahia e ia.Pānuitia katoatia → i te $100 ki te wheropetiWheroHe peti utu ōrite ki ngā tau whero e 18. Wīwī: rouge.Pānuitia katoatia →, ā, ka whakaae ki a ia anō kia n ngā peti i te porowhita, nui rawa: i muri i ngā hinga n karapīpiti ka whakarere, ka tīmata anō. Tuhia te hua toharite o te porowhita kotahi hei tātai i te n, ā, tātaihia mō ngā peti 5, 6 me te 7 i te wīra Ūropi.

   Whakaaturia te otinga

   Whakautu Ko te toharite ko 100 × (1 − (38/37)ⁿ): −$14.26 mō ngā peti 5, −$17.35 mō te 6, ā, −$20.52 mō te 7. Mā te whakaae ki ngā peti maha ake ka kino ake te porowhita toharite, kaua e pai ake.

   Ka toa te porowhita i te $100 ki te kore e hinga ngā peti n katoa, ā, ka pērā ko te tūpono qⁿ, ko q = 19/37. Ina hinga katoa, ka tāpiri ngā moni peti 100, 200 … 100 × 2ⁿ⁻¹ ki te 100 × (2ⁿ − 1). Nō reira ko te toharite ko 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).

   I konei 2q = 38/37, he paku nui ake i te 1, nō reira ka tipu te (2q)ⁿ me te n, ā, ka heke te toharite: ka hoatu e n = 5 te 1.1426 me te −$14.26; e n = 6 te 1.1735 me te −$17.35; e n = 7 te 1.2052 me te −$20.52. He iti ake te auau o te hinga o te arawhata roa ake, i te 3.6%, 1.8% me te 0.9% o ngā porowhita, engari he rua whakarea te rahi o ia hinga, ā, he nui ake ngā moni ka whakawhiti i te tēpu i ia porowhita, i te 2.70% o ia peti.

   I te wīra Amerika 2q = 40/38, ā, ko ngā tatauranga ko −$29.24, −$36.04 me te −$43.20 (Taylor, 2021). I tētahi kēmu kāore he korewīraKoreTe kōhao kākāriki kotahi kāore e kapia e tētahi peti utu ōrite, tatini, tīwae rānei; te pūtake o te painga o te whare. Arā anō: kore kotahi, 0.Pānuitia katoatia → 2q = 1, ā, ko te toharite o ia porowhita ko te kore tonu, ahakoa te roa o te arawhata.

2. Ka whakaruatia e te “Martingale nui” te moni peti i muri i te hinga, ka tāpiri i te waemoniWaeTe moni peti taketake e mahi ai tētahi pūnaha, tētahi kaitākaro rānei. Ka whakaahuatia ngā kauneke ki ngā wae kia hāngai ai ki ia tēpu.Pānuitia katoatia → kotahi, nō reira ko ngā moni peti ko 1, 3, 7, 15, 31 … E hia tō piki ina puta te toa tuatahi i te peti tua-1, tua-2, tua-3, tua-k rānei? I te wae $10 me te tepe nui $500, kia hia te auau o te pakaru o te porowhita, ā, he aha te utu o te porowhita i te toharite?

   Whakaaturia te otinga

   Whakautu Ko te toa i te peti tua-k ka waiho koe kia k ngā wae kua piki: +1, +2, +3, ā, pērā tonu. I te $10 me te $500 ka pakaru te porowhita i muri i ngā hinga e rima, 3.57% o ngā porowhita, mō te −$570. I te toharite ka ngaro i te porowhita te $2.32 i ngā $85.73 i petihia: 2.70%.

   Ko te moni peti tua-k ko te 2ᵏ − 1 wae: 1, 3, 7, 15, 31. Ka tāpiri ngā moni peti k − 1 tuatahi ki te 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Ka utu te toa i te peti tua-k i te 2ᵏ − 1, nō reira ka mutu te porowhita kua piki (2ᵏ − 1) − (2ᵏ − k − 1) = k wae: kotahi te wae tāpiri mō ia hinga i mua i te toa.

   I te $10 i te wae ko ngā moni peti ko $10, $30, $70, $150 me te $310; ko te mea e whai ake ana, $630, kei runga ake i te tepe nui $500. Ko ngā hinga e rima karapīpiti, (19/37)⁵ = 3.57%, e 10 + 30 + 70 + 150 + 310 = $570 te utu.

   Toharite: ka kawea mai e ngā toa te $10 × k ko te tūpono (19/37)ᵏ⁻¹ × 18/37 mō k = 1 ki te 5, ā, ka tangohia e te pakaru te $570 ko te tūpono 3.57%; tahi −$2.32. Ko te toharite o ngā moni i petihia ko te $85.73 i te porowhita, ā, 2.32/85.73 = 2.70%. Ko te Martingale noa i taua tēpu anō ka pakaru i muri i ngā hinga e ono, 1.83% o ngā porowhita, mō te −$630, ā, ka ngaro te $1.74 i ngā $64.20 i petihia: 2.70% anō.

3. Ka whakamahere tētahi kaitākaro i te Martingale ki te whero me te peti tuatahi $1 i tētahi tēpu ko tōna tepe ko te $1,000, nō reira e tekau ngā peti ka taea: $1, $2, $4, ā, pērā tonu ki te $512. Kia hia te auau ka ngaro i te porowhita ngā tekau katoa, kia pēhea te roa i te toharite kia pērā, ā, kia hia te auau ka mutu pērā tētahi pō o ngā peti 200?

   Whakaaturia te otinga

   Whakautu Ka ngaro i te porowhita ngā tekau katoa tata ki te kotahi i ngā 784, ā, e $1,023 te utu. Tata ki ngā peti 1,600 tēnā i te toharite, ā, ka tūtaki tētahi pō o ngā peti 200 ki te pakitara tata ki te 11% o te wā. Ka ngaro tonu i te porowhita te $0.31 i te toharite i ngā $11.31 i petihia: 2.70%.

   Ka hinga te peti ki te whero ko te tūpono 19/37, kei roto te kore. Tekau karapīpiti: (19/37)¹⁰ ≈ 0.00128, tata ki te 1 i ngā 784. Ka tāpiri ngā moni peti 1 + 2 + 4 + … + 512 ki te $1,023.

   E 2.05 ngā peti o te porowhita i te toharite, nō reira tata ki ngā peti 1,600 mō ngā porowhita 784. Ki te tatau i ia hurihanga, ka puta tētahi raupapa o ngā hinga tekau i tētahi wāhi o ngā peti 200 i te 11.3% o te wā.

   I waenga i ngā aituā ka toa te kaitākaro i te $1 i ia porowhita: tata ki te $783 i mua i te mea kino, kātahi −$1,023, tata ki te −$240 puta noa i te wā katoa. I ia porowhita: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Ngā moni i petihia i ia porowhita: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Ā, −0.31/11.31 = −2.70%.

   Whakatauritea ki te peti tuatahi $10 i te tepe $500, e ono ngā peti: ka tae mai te pakitara kotahi i ngā porowhita 55, ā, ka tūtaki tētahi pō o ngā peti 200 ki a ia i te 84% o te wā. Mā te tepe teitei ake ka onge ake, ka nui ake te hinganga, engari kāore e iti iho te utu.