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. Car
3. Car
4. Goat
5. Goat
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 336 times and lost 664 times.

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