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 3 times and lost 7 times.

1. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
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 336 times and lost 664 times.

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