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 4 times and lost 6 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
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 328 times and lost 672 times.

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