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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
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 311 times and lost 689 times.

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