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. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Car
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 370 times and lost 630 times.

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

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 103 times and lost 897 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 33 times and lost 67 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;
	}
}