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. Goat
2. Goat
3. Goat
4. Car
5. Car
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 334 times and lost 666 times.

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