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. Car
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
10. Goat

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 341 times and lost 659 times.

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