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 6 times and lost 4 times.

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

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 329 times and lost 671 times.

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