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. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
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 321 times and lost 679 times.

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