Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 1 time and lost 9 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 332 times and lost 668 times.

1. Goat
2. Car
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Goat
14. Goat
15. Car
16. Goat
17. Goat
18. Goat
19. Car
20. Goat
21. Goat
22. Goat
23. Goat
24. Goat
25. Car
26. Goat
27. Goat
28. Goat
29. Car
30. Goat
31. Car
32. Car
33. Goat
34. Car
35. Goat
36. Goat
37. Goat
38. Car
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Car
51. Car
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Car
58. Goat
59. Goat
60. Car
61. Car
62. Goat
63. Goat
64. Car
65. Goat
66. Car
67. Goat
68. Car
69. Car
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Car
78. Goat
79. Goat
80. Goat
81. Car
82. Goat
83. Goat
84. Goat
85. Goat
86. Goat
87. Goat
88. Car
89. Car
90. Car
91. Goat
92. Goat
93. Car
94. Goat
95. Goat
96. Goat
97. Goat
98. Car
99. Goat
100. Goat
101. Car
102. Goat
103. Car
104. Goat
105. Goat
106. Goat
107. Car
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Car
117. Car
118. Car
119. Car
120. Goat
121. Goat
122. Car
123. Goat
124. Goat
125. Car
126. Car
127. Car
128. Goat
129. Goat
130. Goat
131. Car
132. Goat
133. Car
134. Goat
135. Goat
136. Car
137. Goat
138. Goat
139. Goat
140. Goat
141. Car
142. Goat
143. Goat
144. Goat
145. Goat
146. Car
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Car
155. Goat
156. Car
157. Goat
158. Goat
159. Goat
160. Goat
161. Car
162. Goat
163. Goat
164. Car
165. Car
166. Car
167. Goat
168. Goat
169. Car
170. Goat
171. Goat
172. Goat
173. Car
174. Car
175. Car
176. Goat
177. Goat
178. Car
179. Goat
180. Goat
181. Goat
182. Car
183. Car
184. Goat
185. Goat
186. Car
187. Goat
188. Goat
189. Car
190. Goat
191. Goat
192. Car
193. Goat
194. Goat
195. Car
196. Goat
197. Goat
198. Car
199. Goat
200. Goat
201. Goat
202. Goat
203. Car
204. Goat
205. Goat
206. Car
207. Goat
208. Goat
209. Car
210. Car
211. Goat
212. Car
213. Goat
214. Goat
215. Car
216. Goat
217. Car
218. Goat
219. Goat
220. Car
221. Goat
222. Goat
223. Car
224. Goat
225. Car
226. Goat
227. Goat
228. Goat
229. Goat
230. Car
231. Car
232. Goat
233. Car
234. Goat
235. Goat
236. Car
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Car
244. Car
245. Goat
246. Car
247. Car
248. Goat
249. Car
250. Goat
251. Car
252. Goat
253. Goat
254. Car
255. Goat
256. Goat
257. Goat
258. Car
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Car
265. Car
266. Goat
267. Goat
268. Car
269. Goat
270. Goat
271. Goat
272. Car
273. Goat
274. Car
275. Goat
276. Goat
277. Goat
278. Goat
279. Goat
280. Car
281. Goat
282. Goat
283. Goat
284. Car
285. Goat
286. Car
287. Goat
288. Goat
289. Car
290. Goat
291. Goat
292. Car
293. Goat
294. Goat
295. Car
296. Goat
297. Goat
298. Goat
299. Goat
300. Goat
301. Car
302. Car
303. Goat
304. Goat
305. Car
306. Car
307. Car
308. Goat
309. Goat
310. Car
311. Car
312. Car
313. Goat
314. Goat
315. Goat
316. Goat
317. Goat
318. Goat
319. Goat
320. Car
321. Car
322. Goat
323. Car
324. Car
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Car
331. Goat
332. Goat
333. Car
334. Goat
335. Car
336. Goat
337. Goat
338. Car
339. Goat
340. Goat
341. Car
342. Car
343. Goat
344. Car
345. Goat
346. Goat
347. Goat
348. Car
349. Goat
350. Car
351. Car
352. Goat
353. Goat
354. Goat
355. Goat
356. Car
357. Goat
358. Car
359. Car
360. Car
361. Goat
362. Goat
363. Goat
364. Goat
365. Goat
366. Car
367. Goat
368. Car
369. Goat
370. Goat
371. Goat
372. Car
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Car
382. Car
383. Goat
384. Goat
385. Goat
386. Goat
387. Goat
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Goat
401. Car
402. Car
403. Goat
404. Car
405. Goat
406. Goat
407. Goat
408. Goat
409. Car
410. Goat
411. Goat
412. Goat
413. Car
414. Car
415. Goat
416. Goat
417. Car
418. Goat
419. Car
420. Goat
421. Car
422. Goat
423. Goat
424. Car
425. Car
426. Car
427. Car
428. Goat
429. Car
430. Goat
431. Goat
432. Car
433. Car
434. Car
435. Goat
436. Car
437. Goat
438. Goat
439. Car
440. Goat
441. Car
442. Goat
443. Goat
444. Car
445. Car
446. Goat
447. Car
448. Car
449. Car
450. Car
451. Goat
452. Goat
453. Car
454. Goat
455. Goat
456. Car
457. Goat
458. Car
459. Goat
460. Goat
461. Goat
462. Goat
463. Goat
464. Goat
465. Goat
466. Goat
467. Car
468. Goat
469. Goat
470. Car
471. Goat
472. Goat
473. Goat
474. Goat
475. Car
476. Car
477. Goat
478. Goat
479. Car
480. Car
481. Goat
482. Goat
483. Goat
484. Goat
485. Goat
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Goat
492. Goat
493. Car
494. Car
495. Goat
496. Car
497. Goat
498. Car
499. Goat
500. Goat
501. Goat
502. Goat
503. Goat
504. Goat
505. Goat
506. Goat
507. Goat
508. Goat
509. Goat
510. Goat
511. Car
512. Car
513. Goat
514. Goat
515. Goat
516. Car
517. Car
518. Goat
519. Goat
520. Goat
521. Goat
522. Car
523. Car
524. Goat
525. Goat
526. Goat
527. Car
528. Car
529. Goat
530. Car
531. Goat
532. Goat
533. Car
534. Car
535. Goat
536. Goat
537. Goat
538. Car
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Car
545. Goat
546. Car
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Goat
553. Car
554. Goat
555. Goat
556. Car
557. Car
558. Goat
559. Goat
560. Car
561. Goat
562. Car
563. Goat
564. Goat
565. Goat
566. Car
567. Goat
568. Goat
569. Goat
570. Car
571. Car
572. Goat
573. Goat
574. Goat
575. Car
576. Goat
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Car
586. Goat
587. Goat
588. Goat
589. Car
590. Goat
591. Car
592. Goat
593. Car
594. Car
595. Goat
596. Goat
597. Goat
598. Car
599. Car
600. Goat
601. Goat
602. Goat
603. Goat
604. Car
605. Goat
606. Goat
607. Goat
608. Goat
609. Goat
610. Car
611. Goat
612. Goat
613. Goat
614. Goat
615. Car
616. Goat
617. Goat
618. Car
619. Car
620. Car
621. Goat
622. Goat
623. Goat
624. Car
625. Goat
626. Goat
627. Goat
628. Car
629. Car
630. Car
631. Goat
632. Car
633. Goat
634. Car
635. Car
636. Goat
637. Goat
638. Car
639. Car
640. Car
641. Goat
642. Goat
643. Car
644. Goat
645. Goat
646. Goat
647. Car
648. Goat
649. Goat
650. Goat
651. Goat
652. Goat
653. Car
654. Goat
655. Car
656. Car
657. Goat
658. Goat
659. Goat
660. Goat
661. Goat
662. Goat
663. Car
664. Car
665. Goat
666. Goat
667. Goat
668. Car
669. Goat
670. Goat
671. Goat
672. Goat
673. Goat
674. Goat
675. Goat
676. Goat
677. Car
678. Goat
679. Car
680. Car
681. Car
682. Goat
683. Goat
684. Goat
685. Goat
686. Car
687. Goat
688. Car
689. Goat
690. Car
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Car
698. Goat
699. Car
700. Car
701. Goat
702. Goat
703. Goat
704. Car
705. Car
706. Goat
707. Car
708. Goat
709. Goat
710. Goat
711. Goat
712. Car
713. Goat
714. Goat
715. Goat
716. Car
717. Goat
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Goat
725. Car
726. Car
727. Goat
728. Goat
729. Goat
730. Car
731. Goat
732. Car
733. Car
734. Goat
735. Car
736. Car
737. Goat
738. Goat
739. Goat
740. Goat
741. Goat
742. Goat
743. Car
744. Car
745. Goat
746. Goat
747. Car
748. Goat
749. Car
750. Goat
751. Car
752. Goat
753. Car
754. Goat
755. Goat
756. Car
757. Goat
758. Goat
759. Goat
760. Car
761. Car
762. Goat
763. Car
764. Goat
765. Goat
766. Car
767. Car
768. Car
769. Car
770. Goat
771. Goat
772. Goat
773. Car
774. Car
775. Car
776. Goat
777. Goat
778. Goat
779. Car
780. Goat
781. Goat
782. Goat
783. Car
784. Car
785. Goat
786. Goat
787. Goat
788. Car
789. Goat
790. Goat
791. Goat
792. Car
793. Car
794. Goat
795. Car
796. Goat
797. Goat
798. Car
799. Goat
800. Goat
801. Goat
802. Goat
803. Car
804. Goat
805. Goat
806. Goat
807. Car
808. Goat
809. Goat
810. Goat
811. Goat
812. Goat
813. Car
814. Goat
815. Goat
816. Goat
817. Car
818. Goat
819. Car
820. Car
821. Goat
822. Goat
823. Goat
824. Car
825. Car
826. Goat
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Goat
836. Goat
837. Car
838. Goat
839. Goat
840. Goat
841. Car
842. Goat
843. Car
844. Goat
845. Car
846. Car
847. Goat
848. Goat
849. Goat
850. Goat
851. Car
852. Goat
853. Car
854. Goat
855. Car
856. Car
857. Car
858. Car
859. Car
860. Car
861. Car
862. Goat
863. Car
864. Car
865. Car
866. Goat
867. Goat
868. Car
869. Goat
870. Goat
871. Goat
872. Goat
873. Goat
874. Goat
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Car
881. Goat
882. Car
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Goat
889. Goat
890. Goat
891. Goat
892. Goat
893. Car
894. Goat
895. Goat
896. Goat
897. Car
898. Car
899. Goat
900. Car
901. Goat
902. Goat
903. Car
904. Car
905. Goat
906. Car
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Goat
916. Goat
917. Car
918. Goat
919. Goat
920. Goat
921. Car
922. Goat
923. Goat
924. Goat
925. Goat
926. Goat
927. Goat
928. Goat
929. Car
930. Car
931. Car
932. Goat
933. Car
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Car
940. Car
941. Goat
942. Goat
943. Goat
944. Goat
945. Car
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Goat
952. Goat
953. Car
954. Car
955. Goat
956. Goat
957. Goat
958. Goat
959. Car
960. Goat
961. Car
962. Goat
963. Goat
964. Goat
965. Goat
966. Goat
967. Car
968. Goat
969. Goat
970. Goat
971. Goat
972. Car
973. Car
974. Goat
975. Car
976. Car
977. Car
978. Goat
979. Goat
980. Goat
981. Goat
982. Car
983. Goat
984. Car
985. Goat
986. Car
987. Goat
988. Car
989. Car
990. Goat
991. Goat
992. Goat
993. Goat
994. Goat
995. Goat
996. Car
997. Car
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 87 times and lost 913 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 26 times and lost 74 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}