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. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
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 340 times and lost 660 times.

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