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. Goat
6. Car
7. Car
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 322 times and lost 678 times.

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