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. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
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 346 times and lost 654 times.

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